interface network models for complex urban infrastructure systems

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Interface Network Models for Complex Urban Infrastructure

Systems

James Winkler1, Leonardo Duenas-Osorio2, A.M.ASCE

Robert Stein3 Devika Subramanian4

ABSTRACT

The reliability assessment of infrastructure systems providing power, natural gas, and

potable water is an integral part of societal preparedness to unforeseen hazards. The topo-

logical properties of interface networks connecting electric substations to water pumping

stations and natural gas compressors have received little attention despite the key role these

connections play in operation and failure propagation. This work introduces a performance

assessment methodology for coupled infrastructures that links physical fragility modeling

with the topology of realistic and ideal connecting interfaces. Distinct interfaces based on

features such as betweenness, clustering, vertex degree, and Euclidean distance are assessed

regarding their role in connecting utility systems and propagating failures from random

and hurricane events in Harris County, Texas, United States. The interface minimizing the

Euclidean distance between electric substations and other utility nodes exhibits a slow per-

formance decline as random failures increase, and retains the greatest functionality under

hurricane events compared to alternative interfaces, although it suffers from limited efficiency

and controllability during normal operation. A convenient hybrid interface using both be-

tweenness and distance features shows adequate performance during normal operation while

1Dept. of Chemical Engineering, Texas A&M University, College Station, Texas 77843 U.S.A, E-mail:[email protected]

2Assistant Professor, Department of Civil and Environmental Engineering, Rice University, 6100 MainStreet, Houston, Texas 77005, U.S.A. Corresponding author: Tel.: +1-713-348-5292; fax: +1-713-348-5268;Email: [email protected]

3Lena Gohlman Fox Professor, Department of Political Science, Rice University, 6100 Main Street, Hous-ton, Texas 77005. Email: [email protected]

4Professor, Departments of Computer Science and Electrical Engineering, Rice University, 6100 MainStreet, Houston, Texas 77005. Email: [email protected]

1

Journal of Infrastructure Systems. Submitted March 24, 2010; accepted May 17, 2011; posted ahead of print May 18, 2011. doi:10.1061/(ASCE)IS.1943-555X.0000068

Copyright 2011 by the American Society of Civil Engineers

http://www.editorialmanager.com/jrniseng/download.aspx?id=70782&guid=f60bebd7-d98e-40f6-8ce4-64d9e32e920c&scheme=1

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exhibiting tolerance to random failures and sufficient performance at increasing hurricane

event levels. These findings provide utility owners and operators with new simple yet ade-

quate strategies focused at the interface across complex systems to enhance routine operation

and reduce the probability of widespread interdependent failures following disruptive events.

Subject Headings:

Hurricanes, Simulation models, Electric transmission, Natural Gas, Water distribution sys-

tems, System reliability, Networks, Interactions

INTRODUCTION

The infrastructure of the Gulf Coast of the United States endures significant economic,

social, and physical disruption due to hurricanes, along with other random events such as

utility system failure from aging, heat, collisions, and animals. These events adversely impact

the functionality of networks that supply critical services, such as power, water, and natural

gas to consumers. However, lifeline systems must be sufficiently reliable and governable

to ensure those persons living and working in vulnerable regions or remaining in disaster

afflicted areas have access to vital utility services. The resulting complex interdependent

system from coupled lifeline systems responds distinctly to disruptions due to unique fail-

ure propagation patterns through a poorly characterized web of interconnections linking,

for example, electric substations to water pumping stations and natural gas compressors.

Topological and physical properties of these interconnections, denoted here as interface net-

works, can play an initial role explaining how failures are transmitted across infrastructure

systems. While sophisticated damage prediction models exist for many types of individual

infrastructure systems, the relationship between these systems and the topological properties

of their interface networks has garnered little attention. This study provides a new simple

yet adequate approach to estimate the performance of complex urban infrastructures subject

to disruptive events by monitoring the response of their interface networks as a surrogate

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of the entire set of interdependent systems. Interface analysis combines physical infrastruc-

ture damage modeling from hurricane and random failures with the topological features of

realistic and hypothetical coupling patterns across lifeline systems.

The complex emergent behavior of lifeline systems arises from the presence of several

interconnected networks, each responding uniquely to external events, internal failures, and

operation errors (Haimes and Jiang 2001; Rinaldi et al. 2001; McDaniels et al. 2007).

Models for interdependent systems that employ graph theoretic and flow based methods use

different levels of sophistication to determine coupled system performance when subjected

to systematic element removals (e.g. an ordered list of removed substations), random events

such as equipment failures, or predefined hazard events (Newman et al. 2005; Apostolakis

and Lemon 2005; Duenas-Osorio et al. 2007; Lee et al. 2007; Min et al. 2007a; Svendsen and

Wolthusen 2007; Rosato et al. 2008; Ouyang et al. 2009). These network based methods

have also been coupled with structural reliability theory to assess the seismic performance

of single infrastructure systems (Hwang et al. 1998; Scawthorn et al. 2006; Shinozuka et al.

2007; Adachi and Ellingwood 2009), or co-located interdependent infrastructures (Duenas-

Osorio et al. 2007; Adachi and Ellingwood 2008). However, few studies have assessed the

performance of the distinct interface networks that couple lifeline systems when subjected to

external disruptions, and how the performance of interface networks may contain information

about the entire set of interdependent systems. The National Infrastructure Simulation and

Analysis Center (NISAC) has explored modeling of interdependent infrastructure and social

networks (Skanata and Byrd 2007; Min et al. 2007b) but without coupling these models to

simulations of natural hazards and component vulnerability. Hence, a new methodology that

incorporates network theory and physical reliability perspectives to approximate complex

interdependent system performance is developed through interface networks, specially to

address disruptions different from earthquake events, such as hurricanes. Random component

failures are also considered to keep a reference point with previous studies.

Among all infrastructure systems, the impact of disruptions upon power systems has

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been a topic of heightened interest due to its socio-economic importance. For instance, the

response of power distribution and transmission systems to storms, hurricanes, earthquakes,

lightning strikes, wind downbursts, and terrorism has been measured using customer quality

of service metrics through a wide variety of modeling techniques. These approaches typically

include analytical and numerical simulation methods based on network element failure rates

for cases where system topology is available (Brown et al. 1997; Chassin and Posse 2005;

Balijepalli et al. 2005; Xie and Li 2009) and empirically-based statistical outage prediction

models when system topology is unavailable (Davidson et al. 2003; Liu et al. 2005; Reed

2008; Han et al. 2009). In addition, power network damage models based on element

fragility, or failure probability of network elements given a hazard intensity level at their

sites, are also used to estimate system-level physical response to extreme events in reliability

and risk based frameworks (Shinozuka et al. 2007; Duenas-Osorio and Vemuru 2009). From

these different perspectives, the physical fragility damage approach is sufficiently flexible

to account for spatial heterogeneity of extreme hazards, uncertainty in network element

fragilities, equipment and reconfiguration network mitigation effects, and the addition of

network elements and operational constraints typical of evolving infrastructures. Therefore,

element based fragility modeling techniques are chosen as a framework to assess the impact

of hurricanes and random failures upon the elements found in infrastructure systems.

Joint network topology and vulnerability analyses for individual systems represent an

additional emerging trend based on graph theory and statistical physics to determine how

topology (the organization of connections between nodes) is linked to system performance

under both normal and abnormal operating conditions. Topological analyses typically focus

on expanding original network centrality ideas (Freeman 1977; Brandes 2001) to technologi-

cal network criticality concepts, and identifying vulnerable elements, determining probability

distributions of link and node degrees, and establishing correlations between network ele-

ments (Newman 2003; Boccaletti et al. 2006; Rosas-Casals et al. 2007). Also, typical single

network topological studies look at the response to random or targeted attacks (Barabasi

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and Albert 2002; Albert et al. 2004; Crucitti et al. 2004; Holmgren 2006; Rosas-Casals

et al. 2007; Sole et al. 2003), and the effects of overloads and error cascades on network

operation (Carreras et al. 2004; Dobson et al. 2006; Pepyne 2007; Simonsen et al. 2008;

Duenas-Osorio and Vemuru 2009; Arianos et al. 2009).

In the context of the complex interacting systems of this study, graph theoretic con-

cepts related to bipartite graphs, or networks whose nodes can be divided into disjoint sets

(Zhou et al. 2007; Peruani et al. 2007; Mukherjee et al. 2009), are adopted to represent

the structure of interface networks that approximately capture how interconnections affect

failure propagation between distinct lifeline systems. A surrogate metric of the ability of

interface networks to connect electric substations to load points of other utility systems,

termed interface efficiency, is also adapted here from efficiency metrics for individual net-

works (Boccaletti et al. 2006). This and other aspects of individual and interface network

topology must be considered alongside the physical fragility of network elements to provide

new insight into the properties of interdependent utility systems to withstand disruptive

events.

In sum, this work explores the topological and physical properties of realistic and ideal

interface networks connecting elements of the power, water, and gas networks of Harris

County, Texas, United States, and reveal how these interfaces, acting as surrogates of the

entire set of complex coupled systems, provide insights about urban infrastructure perfor-

mance in the context of random failures and hurricane events. Fundamental properties of

the infrastructure systems under consideration are provided in the next Section. Then,

the paper presents element based physical fragility and damage models introduced recently

by Winkler et al. (2010), but expanded here to study the performance of coupled infras-

tructures. The methodology for constructing the electrical-utility interface from topological

properties is then defined, followed by topological analysis of the synthesized networks con-

necting the electrical-water and electrical-gas networks within Harris County. The expanded

hurricane and random damage models are subsequently applied to determine how these

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interface networks respond to damage. Then, the paper analyzes results of interface perfor-

mance to ultimately inform design and retrofit recommendations in practice. Finally, the

paper presents major conclusions concerning the properties of coupled infrastructure systems

under disruptions.

DEFINITION OF INTERDEPENDENT INFRASTRUCTURE SYSTEMS

Let GP (VP , EP ) indicate an annotated, simple, undirected graph representing the power

transmission network with an electrical generation and substation node set VP , and a trans-

mission line (link) set EP connecting nodes within VP (Diestel 2006). The choice of an

undirected electrical network is justified by the frequency of flow reversals during outage

events (Chassin and Posse 2005). Since interface networks control the passage of power from

the electrical network to the dependent water (w) or gas (g) utilities, they are represented by

annotated, bipartite, undirected graphs, denoted by GIu(VIu , Vu, Eu) where u = {w, g}, VIu

is a subset of electrical substation nodes that supply power to water or gas demand nodes

through interface networks (VIu ⊂ VP ), Vu are interface water or gas demand node sets,

and Eu are edge sets representing power transmission lines to serve interface demand nodes.

Actual flows of water and gas in their respective infrastructure networks are not modeled in

order to focus on the interface topologies that provide linkages from power supply to util-

ity demand nodes. These interface networks are constructed according to topological and

geographical considerations, and serve as a first approximation of infrastructure coupling

topology so that their effect on interdependent system reliability may be investigated. The

combination of interface network vertices and edges along with the rest of the power system

elements is referred to as the Harris County complex system in this study. Properties of the

power network are presented in Table 1, where order and size denote the number of nodes

and edges, respectively, and vertex degree d(v) indicates the number of links per node v.

The demand nodes of the potable water network obtained from the City of Houston are used

to represent the water supply system for the entire Harris county, as most county pump-

ing stations (66 in total) are located within Houston (Kathy Chan, City of Houston Public

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Works and Engineering Department 2009). The Harris County power and natural gas trans-

mission network connectivity data are obtained from Platts (Platts 2009); there are only 7

gas demand nodes (compressors) within the county area. A geographical representation of

the Harris County complex system without interface links is introduced in Figure 1.

Failures in the power transmission network affect the level of service the power grid can

deliver immediately, resulting in an instantaneous reduction in power supply, and eventual

water or gas service efficiency reductions. The electrical transmission infrastructure is also

physically more vulnerable to wind damage compared to water and gas infrastructure since

their associated elements, such as water pumping and gas compression stations connected

by underground pipelines, are located underground (Quanta Technology 2009). Hence, elec-

trical network reliability (only in a connectivity sense) coupled with fragility assessment of

vulnerable water or gas interface elements is a major determinant of interface performance

under disruptive conditions in this study.

INFRASTRUCTURE AND ELEMENT DAMAGE MODELING

The joint transmission-distribution element damage model developed by Winkler et al.

(2010) for evaluating hurricane impact upon power networks is adapted here to include fail-

ure estimates of the water and gas nodes. The original power system damage model uses

several fragility relationships, to determine element and system level residual performance

via Monte Carlo simulation. Hurricane fragility curves along with engineering design provi-

sions provide estimates of physical substation damage and disconnection (American Society

of Civil Engineers 2008; Personal Communication with Frank Lavelle (Applied Research As-

sociates) 2008; Vickery et al. 2006), power transmission line failure (American Society of

Civil Engineers 2008; American Society of Civil Engineers 2010), and abnormal current flow

through topological approximations (Albert et al. 2004; Kinney et al. 2005; Duenas-Osorio

and Vemuru 2009). This power system reliability model is upgraded here to include fragility

functions of water and natural gas network nodes and produce a model capable of assessing

the performance of the Harris County complex system at the transmission level. In general,

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infrastructure element fragility models also account for local terrain features (e.g., rough-

ness), element physical properties, and other local conditions, such as flying debris potential

in addition to the wind speed at their sites. Random failure simulation is also employed

to evaluate how interface networks respond to increasing levels of node and edge failures,

without the need for fragility models.

Simulation of Random Failures

Random failures and their effect on the interfaces connecting utility networks are simu-

lated by randomly removing electrical substations, transmission lines and water or gas utility

elements from the Harris complex system. Edges incident upon a randomly removed node are

also removed from the networks. Any power system elements separated from generation ca-

pacity are also assumed to have failed. The total number of direct and disconnection random

failures affecting an interface is given by the fraction of nodes (fn) and edges (fe) removed

from the entire complex system. Topological properties and performance of the randomly

damaged interfaces are averaged over fifty iterations per removed element fraction.

Simulation of Hurricane Hazards

The hurricane performance of the Harris complex system is tested using a series of

simulated events, with return periods ranging from 10 to 1 000 years, generated with the

HAZUS-MH 3 software package from the Federal Emergency Management (FEMA) (Vick-

ery and Twisdale 1995; Vickery et al. 2006; Federal Emergency Management Agency 2008).

Wind gusts are predicted on a census tract level of geographical granularity. Each census

tract optimally contains an average of 4 000 people, leading to large variations in their geo-

graphic coverage (United States Census Bureau 2005). Land cover data for Harris County,

used to incorporate the effect of terrain and ground roughness such as forests, suburban

dwellings, and open fields on wind gust strength, is obtained from the Multi-Resolution

Land Characteristics Consortium (Liu et al. 2005; Winkler et al. 2010; Multi-Resolution

Land Characteristics Consortium 2001). This set of representative wind gusts at the sites of

infrastructure network elements is used to establish their probability of failure and determine

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the extent of damage at the interface network and associated complex system levels.

Hurricane-induced Failure of the Harris County Complex System

In addition to the power network performance assessment methods already considered

by the Winkler et al. (2010) model, this work considers the wind fragility of above ground

potable water pumping stations and natural gas compressors. These structures are assumed

to be relatively small one story commercial buildings (Tobin 2007; Chapin and Dewberry

2006). The water and gas pipe lines between utility nodes are not included in fragility anal-

yses, since they are typically underground and impervious to wind damage, and are not

captured by interface bipartite network models. However, regions experiencing widespread

wind-induced surge hazards should incorporate flood damage modeling (Abdalla et al. 2007).

As with the power system damage model, pertinent log-normal fragility curves similar to

Equation (1) are selected from HAZUS-MH 3 to assess the probability of reaching or exceed-

ing a moderate level of wind damage to the jth water or natural gas network nodes:

P (Failj|xj) =

∫ ∞xj

1√2πσ

exp

(−(ln(xj)− µ)2

2σ2

)dx (1)

where xj is the wind gust speed at the jth element site, and µ and σ are the logarithmic

mean and standard deviation of the curve. The probability of direct damage to water or gas

elements is low overall due to the required design standards applied to these installations

(Sanks et al. 2008). Specifically, water and gas utility elements are assumed to have fragili-

ties equivalent to that of single story steel commercial installations, with median fragilities

in the range of 60-72 ms−1 depending on local terrain roughness (Federal Emergency Man-

agement Agency 2008). Water and gas utility failures due to disconnections from the power

transmission are expected to be significantly more common than direct damage from hurri-

cane winds (Winkler et al. 2010). Critical water and gas facilities are typically connected to

backup generation capacity to maintain a steady supply of power in case of outages (Sanks

et al. 2008). However, it is assumed in this study that backup generation fuel supplies do not

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last long compared to the length of hurricane-induced outages, as occurred in the aftermath

of Hurricane Ike in 2008 in Harris County (Centerpoint Energy 2008). Edges between the

power and other utility networks are modeled as typical overhead transmission lines. The

probability of transmission line failure is determined by the ratio of wind force on each line

and the rated line strength as a first approximation (Winkler et al. 2010). Fifty Monte Carlo

simulation iterations of the extended multi-system fragility based damage model are applied

to evaluate the impact of each wind hazard scenario on the electrical system and water or

gas demand nodes on their associated interface network connections.

INFRASTRUCTURE INTERFACES

An exploration of diverse realistic and ideal interface topologies provides clues as to how

these coupling networks can be designed or retrofitted to best tolerate extreme events and

other failures. The construction of interface networks, which can also approximate the per-

formance of entire sets of coupled infrastructure systems, consists of selecting a subset of

existing power substation supply nodes VIu and connecting them to water or gas network

demand nodes (Vu) via power transmission lines in Eu. These connections consider topo-

logical and physical characteristics of the electric substation nodes as criteria for inclusion

in VIu , such as vertex degree, clustering, betweenness, and geographical location. Once con-

structed, the topological properties of interface networks reveal how links are distributed

between coupled infrastructure node sets, the number and location of supply nodes critical

to the integrity of the dependent utility networks, and their vulnerability to random failures

and hurricane events.

Electrical Network Topology and Interface Performance Metrics

The construction of realistic and ideal interface networks, as detailed in the following

subsection, first requires the assessment of substation node topological properties within the

electrical transmission networks so that VIu nodes may be preferentially incorporated into

interfaces. Existing nodes in the power transmission and water or gas networks are connected

with synthesized edges based on properties of the electrical substations. Those substations

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critical to electrical network flow are approximately identified using the concept of node

betweenness centrality (the number of shortest paths containing node v in a network) CB(v),

a metric described by Freeman and later implemented by Brandes et al. (Freeman 1977;

Brandes 2001). This metric is widely applied in topological analyses of power transmission

networks (Albert et al. 2004; Kinney et al. 2005; Duenas-Osorio and Vemuru 2009) to

determine those nodes traversed often by flow paths between all nodes within the power

network. Equation (2) describes the computation of CB for all nodes in VP , where σij(v)

is the number of shortest paths between nodes i and j containing node v (subject to the

restrictions that i 6= j and i, j ∈ VP ) and σij is the total number of shortest paths between

i and j.

CB(v) =∑

i6=j,i,j∈VP

σij(v)

σij(2)

Central electric substation nodes (large CB) are likely to become supply nodes of the interface

network sets VIu ; their removal from the interface due to random failures or hurricane damage

seriously impairs network efficiency. Interface nodes may alternatively be selected from the

power grid according to their local importance, as shown by involvement of substations

in cycles of length 3 quantified by their local clustering coefficient C(v) (Newman 2003;

Boccaletti et al. 2006; Lind et al. 2005) defined in Equation (3):

Ci =2ei

ki(ki − 1)(3)

where the clustering coefficient of node i is computed from the number of edges shared by

neighbors of node i (ei) normalized by the total number of connections possible between ki

nodes. Substations with many connections in the electrical network can also be incorporated

into interfaces through the use of node degree d(v) rankings. In particular, the degree d(si) of

electrical substations si ∈ VIp is used to assess their relative importance within an interface

in terms of the availability of main feeders to deliver power. This degree-based ranking

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procedure is often applied to other types of bipartite networks (Newman 2003; Guillaume

and Latapy 2006; Peruani et al. 2007; Mukherjee et al. 2009). Since most power grid

topological properties are influenced by the probability distribution of substation degrees,

their study has revealed they match the exponentially decaying node degree distribution

generated by Equation (4) (Rosas-Casals et al. 2007; Barabasi and Albert 2002):

P (k > k′) =

∫ ∞k′

P (k)dk ≈ exp(−k′/γ) (4)

where P (k > k′) represents the proportion of substation nodes with degree k > k′, P (k)

the probability density of substation nodes with degree k, and γ a constant affecting the

slope of the exponential decay. This general equation may be applied to the power sets VIu ,

since these also display an exponential decay in vertex degree for all synthesized interfaces

as a function of their incident interface edges Eu. The node degree distribution can also be

linked to a theoretical random attack tolerance using the critical fraction of removable nodes

(pc) that marks the threshold before networks decompose into small, unconnected clusters

from increasing levels of random failures (Rosas-Casals et al. 2007; Cohen et al. 2000).

The pc parameter connects the topological properties of a network to expected connectivity

reliability under random attack. To explore interface connectivity reliability when subject to

random failures, Equation (5) is used to compute the pc parameter using the corresponding

γ for the interface substation supply set VIu of each synthesized electric-water or electric-gas

interface network:

pc = 1− 1

2γ − 1. (5)

The size of the largest resulting connected cluster (LCC) from interface construction also pro-

vides initial insight into the overall level of connectivity within an interface, since interfaces

may contain from one to multiple clusters. While these fundamental topological proper-

ties provide a holistic assessment of interface connectivity, more specialized techniques such

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as network efficiency (Boccaletti et al. 2006) are needed to better approximate interface

performance from topology.

The concept of efficiency provides one method to describe interface performance explicitly

in terms of path lengths separating water or gas demand nodes from electrical substation

supply nodes. Boccaletti et al. (2006) utilize relational distance between nodes to define

the efficiency E(G) of a network G, with 0 ≤ E(G) ≤ 1, in terms of node-to-node path

lengths. Since connections between water or gas demand nodes and electrical substations

affect interface reliability, the usual efficiency definition is adapted here to consider only the

relational distance between each node set of the bipartite interface by restricting computation

to the average of supply-demand node path lengths within an interface GIu using Equation

(6):

E(GIu) =1

NsNu

∑si∈VIu ,dj∈Vu

1

RD(si, dj)(6)

where Ns is the number of supply nodes in VIu , Nu the number of water or gas demand

nodes in Vu, and RD(si, dj) is the relational length of the shortest path between electric

supply node si and water or gas demand node dj. This metric is readily applicable to the

undirected networks used in this study. This interface efficiency metric reflects how quickly

it is to reach distant portions of the networks, the ability to share flow and information,

and the operation or controllability of networks in practice. The use of relational distance

as opposed to physical or electrical distance of paths within the power transmission net-

work is deemed appropriate as a first approximation due to the nearly instantaneous travel

time of electricity between supply VIu and demand Vu nodes. Interface networks with short

paths between VIu and Vu exhibit large E(GIu) scores, while networks with very long paths

score poorly under this metric. Damage to the interfaces resulting in the failure of trans-

mission lines or electrical substations will decrease interface efficiency, reducing their ability

to transmit the power necessary to maintain lifeline services. The repeated monitoring of

E(GIu) as hazard intensity increases therefore provides a simple but meaningful approach

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for quantifying the performance change of distinct interfaces before and after random or

hurricane disruptions, and determining if these surrogate interface networks capture entire

interdependent infrastructure systems performance information.

Methodology for Bipartite Interface Graph Construction

Topological properties of the power transmission network are utilized to generate novel

synthetic interface configurations to identify efficient and robust interfaces that tolerate ran-

dom and hurricane-induced failures. Four methods of generating interface connectivities by

maximizing T (si) objective functions for each substation-utility demand node pairing are

defined in Table 2. The objective functions all rely upon topological properties of the elec-

trical supply nodes. These methods are employed to study the relationship of topology with

interface reliability and performance. The mathematical description of the interface gener-

ation process is presented in Algorithm (1). While these networks are theoretical, existing

interface networks seek to optimize many of the same practical constraints, particularly that

of transmission line length to reduce implementation costs. Note that ρ ∈ Z+ is defined as

the number of supply nodes si linked to each demand node dj in the utility system under con-

sideration, although in future studies can be specific to each connection as ρj. The interface

construction schemes search for a subset VIu of power substations in VP , ranked according to

their topological properties in the power transmission network, to supply power connectivity

to interface water or gas demand nodes. One of the simplest and practical approaches is

to link each demand node to power supply nodes that are closest in terms of geographical

proximity via squared Euclidean distances, dist(si, dj)2 (Ouyang et al. 2009; Duenas-Osorio

et al. 2007). Squaring the Euclidean distance disfavors the adoption of impractical long

transmission lines between supply and demand nodes. In this work ρ = 2 is chosen as a first

approximation given that two electrical substations are often connected to water pumping

stations to ensure a redundant and affordable power supply linkage (Sanks et al. 2008); gas

compressors are assumed to be similarly connected.

In addition to Euclidean distance, it is also important to explore other coupling strategies

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that simultaneously maximize different topological criteria and minimize distance between

supply and demand nodes. An alternative to a purely distance based match criterion is to

link water or gas nodes to the ρ closest power substations that maximize the ratio of node

betweenness centrality scores and distance ( CB(si)dist(si,dj)2

). Substations identified as optimal

using this objective function carry high flows within the power transmission network. The

substation node clustering coefficient C(si) is also employed in place of betweenness to pair

water or gas utility nodes to power transmission substations. This strategy ensures local

power supply redundancy. Similarly, ranking based upon substation degree d(si) selects

substations that are connected to several surrounding substations, reflecting the practice of

linking customers to nodes with sufficient power delivery feeders. Along with employing each

ranking criterion in isolation, topologically heterogeneous hybrid interfaces are also created

by pairing supply-demand nodes using two or more connection schemes simultaneously, and

by altering the substation intensity ρ. Schematic forms of the electric-water and electric-gas

interface networks are shown in Figures 2 and 3.

Hybrid interfaces can exploit desirable properties of single interfaces, such as high ef-

ficiency and high resistance to random failures and hurricane events. An example is the

betweenness-distance (B-D) hybrid interface which makes ρ− 1 linkages using the distance

criterion and one additional betweenness-based connection. While this method can be ap-

plied to create clustering or degree based hybrid networks, the B-D combination attempts to

retain the expected high efficiency of the betweenness interface and the topological decentral-

ization of the distance interface while sharing a large portion of the electrical substations in

their VIw sets. In fact, the next section shows that the electric substations of a betweenness

interface are almost entirely subsets of the distance interface, making the B-D hybrid ideal

for retrofits of realistic distance-based interfaces via an added betweenness-based link. This

approach results in ρ = 3 and is denoted by B-D3. Also, interface hybrid designs can be

constructed including a link per parent network criterion, yielding a ρ = 2 and denoted by

B-D2 (Figure 2).

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EVALUATION OF THE COMPLEX SYSTEM INTERFACE NETWORKS

Electric-Water Interface Properties

Key topological properties of the generated electric-water system interfaces are presented

in Table 3. Among non-hybrid interfaces, the distance and clustering approaches represent

extremes in order, substation count, substation degree, edge length, tolerance of random

failures in the form of the critical fraction of removal nodes (pc), and efficiency of the un-

damaged networks. Also, the clustering and betweenness interfaces are topologically closer

overall than the similar degree and distance interfaces. The total length of edges within the

interfaces is of interest due to its role in exposing interface links to wind hazards, since a

smaller total edge length is expected to decrease vulnerability of transmission lines (Winkler

et al. 2010). As expected, the realistic distance interface has the lowest total edge length

and smallest connected cluster, presumably reducing its vulnerability to transmission line

damage and impact on performance changes given its topological decentralization and low

baseline efficiency value. This topological decentralization, resulting from a large number

(82) of conveniently close low degree substations serving demand nodes in the interface, ex-

plains the low overall interface efficiency and low pc, along with low potential for coordinated

operation, and high robustness to changes in graph theoretical efficiency (as shown later in

this subsection). In contrast, the clustering and the betweenness interfaces possess large

connected components, large total edge length, and some high degree substations, yielding

highly efficient topologies that could be easily operated in practice but may be economically

infeasible. These interfaces also have high theoretical random failure tolerance, but at the

expense of possible heightened vulnerability to hurricane disruptions if failure is observed in

their long transmission lines and hub-like substations. The degree interface has a smaller ef-

ficiency and pc than the clustering and betweenness interfaces, and a higher pc and efficiency

than the distance interface, making it a topology likely to be observed in practice. Hybrid

B-D interfaces, such as the retrofit B-D3 and new design B-D2, have intermediate properties

relative to their parent topologies. Hence, hybrid B-D topologies serve as a compromise

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structure that improves the efficiency, random failure tolerance, and practical operation of

distance interfaces, while retaining low edge exposure, small changes to base-line efficiencies,

and feasibility for practical implementation.

The set of power supply nodes VIw generated by each interface construction method is

also of interest due to their correlations. Table 4 indicates the number of electric substa-

tions shared between the connection schemes. The betweenness and clustering connection

schemes share few supply nodes, indicating that the set of nodes tends to be disjoint. Sixty

three supply nodes are shared by practical degree and distance based connection methods.

Interestingly, the betweenness connection scheme is nearly a subset of the distance and de-

gree based interfaces; this commonality is exploited by B-D interfaces to minimize the total

number of substations required for hybridization in retrofit or new designs.

To further explore the topological properties of interface networks, which suggest that

there is a trade-off between interface efficiency, random failure tolerance pc, and imple-

mentability, normalized interface efficiencies are studied as a function of increasing fractions

of node and edge random removals. The normalized scale relative to the intact efficiency

values in Table 3 enables comparisons across interfaces in terms of the rate at which absolute

efficiency decreases upon disruption. The calculated interface network efficiency for each in-

terface shown in Figures 4a-4b reveals significant differences as expected from topological

analysis. While all networks suffer large normalized efficiency declines as fn and fe → 1,

the rate of efficiency decline reveals that the realistic distance and degree interfaces along

with feasible hybrid interfaces achieve an almost linear drop in performance as fe increases

compared to the rapid declines seen for the idealized betweenness and clustering interfaces.

Similar normalized efficiency change trends, although slightly more pronounced, are observed

as fn increases. There are predominantly short edges and homogeneous importance nodes in

the distance, degree and hybrid interfaces (with some departures for the B-D3 interface due

to its ρ = 3) given their topological fragmentation, low average vertex degree, and low pc,

which minimize the overall impact of element removals on interface efficiency changes. In hy-

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brid networks, although most nodes contribute approximately equally to network efficiency

within separate clusters, a few nodes also enhance connectivity and efficiency within and

across clusters. The difference between the interface topological groups (the betweenness

and clustering networks versus the distance and degree networks) is statistically significant

(P-value < 0.05) except at high fractions of node or edge removals.

These betweenness and clustering interfaces also show higher degrading rates of nor-

malized efficiency scores under random node than edge deletion due to their high average

substation degree and important initial LCC size, which translates into a large increase in

distance between nodes and potential loss of controllability. Although not shown in the

normalized efficiency results of Figure 4b, the absolute interface efficiency values at the cor-

responding fn = pc levels oscillate around 0.02 for all interfaces. This observation highlights

the ability of the theoretical pc value to determine a critical state of the interfaces and

associated infrastructures.

Another aspect of complex utility network reliability assessment is hurricane damage tol-

erance. Since random failures do not account for the inherent spatial variability of hurricane

damage arising from local terrain, hazard intensity, and network element fragility factors, the

expanded Winkler et al. model (Winkler et al. 2010) is applied to assess network efficiency

for each interface when exposed to a variety of wind gust scenarios. Interface networks are

exposed to simulated wind fields of hurricanes occurring every 10, 20, 50, 100, 200, 500, and

1 000 years in addition to a HAZUS generated wind field for the 2008 Hurricane Ike. The

mean interface responses in Figure 4c show that the distance interface along with degree and

hybrid interfaces retain greater normalized efficiency than the betweenness and clustering

interfaces at all storm intensities, although the differences are initially more pronounced.

The concave shape of all curves indicates that at low wind speeds, vulnerable edges are more

likely to fail without impacting greatly the normalized efficiency scores of the interfaces, de-

spite the heightened initial fragility and sensitivity to efficiency changes of the betweenness

and clustering structures. As the hazard intensity increases, the rate of network efficiency

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decline increases for all interfaces once the average storm wind speed exceeds approximately

48 m s−1 due to increasing risk of substation failure within the electrical network (Winkler

et al. 2010). The normalized efficiency ranges between {0.55, 0.66} of the distance, degree,

and hybrid interfaces under Hurricane Ike also compares favorably with the 67% retention

of water utility service reported by Harris County residents (Stein 2008), suggesting that the

performance of complex Harris County system is indirectly captured in terms of the physical

fragility and topological features of practical interface models. The standard variation of

the responses across events varies only between 3-5%, so interface performance differences

between the topological groupings are statistically significant (P < 0.05) at all hazard levels.

This joint physical fragility and topology approach suggests that hardening supply nodes

within electric-utility interfaces to better resist hurricane-force winds above the 48 m s−1

threshold may improve interdependent network reliability if the complex system is repeatedly

subjected to strong winds. A factor influencing the resistance of the distance, degree, and

B-D2 interface networks to hurricane disruption is the small total length of transmission lines

connecting water utility nodes to substations and the robustness to efficiency changes from

decentralization.

Electric-Gas Interface Properties

The synthesized electric-gas interfaces are much smaller compared to the corresponding

electric-water interfaces, and lack complex topological structures. The topological properties

of these interfaces in Table 5 indicate that the betweenness and clustering interfaces have

large connected components and higher substation degree than the distance and degree

interfaces, resulting in higher efficiency scores. Given that the majority of power supply

nodes in each interface are only connected to a single gas compressor, the interface network

degree distributions also lacks the distinctive properties of technological networks (Newman

2003), resulting in unimportant pc values or element correlations in the substation sets VIg .

The responses of the electric-gas interfaces to random edge failures are presented in Figure

5a. While small differences in random failures between the electric-gas interfaces are evident,

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the interfaces do not contain enough elements to exhibit distinct efficiency profiles except for

the slightly greater susceptibility of the clustering interface to edge failures. The performance

of the various electric-gas interface networks are statistically insignificant from one another in

their responses to random failures (P-value > 0.05). The principal consideration in studying

small power-gas utility system interfaces is therefore the size of the connected components

from interface construction and the total length of the lines connecting nodes in the two

networks. Hybrid interfaces in this case will not benefit the reliability of the electric-gas

system due to the lack of topological and normalized efficiency loss distinctions between the

potential parent interfaces.

Interface responses to random node failures shown in Figure 5b indicate that all interfaces

exhibit similar but slightly non-linear declines in efficiency as fn → 1 since node removals

include incident edges, in contrast to the linear efficiency profiles under just edge failures.

Hurricane damage responses of interfaces, depicted in Figure 5c, are a product of the node

and edge topological and physical fragility properties coupled with differences in overall

interface edge length. The increased rate of efficiency decline observed in the electric-water

interfaces at wind speeds above 48 m s−1 also occurs in this electric-gas case. In contrast to

the distinct hurricane performance profiles of the electric-water interfaces, the topologically

homogeneous electric-gas interfaces perform similarly under hurricane demand, though the

clustering network does perform less well than the other interfaces at lower wind speeds

(P-value < 0.05). The normalized efficiency predictions for Hurricane Ike, ranging between

{0.62, 0.66}, differ from the 92% total gas utility service retention reported by Harris County

residents (Stein 2008). This discrepancy arises from the small size of the system and the

fact that failure of gas compressors does not directly translate into a loss of system pressure

if other units in the system can compensate or if they can operate for extended time with

their own gas as fuel.

Performance disparities between the power-gas interfaces are not sufficient to justify

significant alteration of practical distance- or degree-based interfaces. Other resources such

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as electricity and potable water are more critical for survival (McDaniels et al. 2007).

DISCUSSION OF RESULTS

The results on interface network performance confirm that when subject to random fail-

ures and hurricane events, interface response is strongly dependent on topology and physical

fragility. The interface network model is shown to capture previously unknown performance

aspects of the coupling mechanisms among utility systems, and their potential relation to

performance of entire infrastructure systems. For instance, topological and physical fea-

tures that have been shown to influence the performance profile of an interface subject to

random or hurricane induced failures include the size of the largest connected component

from interface constructions, the interface substation degree distribution, the edge length

within interfaces, the number and distance between intervening interface elements, and the

physical fragility of all complex infrastructure elements. Interface designs must therefore

compromise between high efficiency, potential controllability (connectivity), and tolerance

to random failures as offered by large connected interfaces, and their vulnerability to hurri-

cane events (exposed transmission lines), sensitivity to efficiency losses, and impracticality

of construction which contrasts with the features of decentralized interfaces.

Interfaces with low baseline efficiency that are composed of low degree nodes and small,

disconnected clusters such as the distance and degree interfaces better tolerate efficiency

changes from increasing levels of random and hurricane disruptions. These interfaces most

closely resemble the actual connection scheme within electric and other utility networks as

most elements are connected based on spatial proximity and power substation feeder avail-

ability. However, the number of supply nodes and decentralized structure in distance and

degree interfaces pose considerable operational control challenges and inefficiencies compared

to more centralized interfaces. High efficiency betweenness and clustering interface networks

require fewer substations and longer edge lengths than low efficiency interfaces, easing control

difficulties, but are in turn more sensitive to element failures resulting in higher normalized

efficiency losses as a function of hazard or element removal intensities. In terms of hybrid

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interface networks, both B-D construction methods produce efficient networks capable of re-

sisting random and hurricane disruptions with performance in between that of their parent

networks. The B-D3 interface in particular offers a promising method to obtain efficient,

controllable, and reliable networks based upon existing interface infrastructure configura-

tions.

There are several recommendations that can be implemented to increase interface re-

sistance to both hurricane damage and random failures. Overhead transmission lines are

especially vulnerable to hurricane wind gusts as their fragility is proportional to line length,

so reducing the total line length of an interface network by employing a distance minimiza-

tion connection scheme will result in fewer utility service disruptions, practical construction,

and reduced sensitivity to efficiency reduction. However, a few long lines are needed to

ease operational control challenges, increase tolerance to random failures, and maintain an

adequate baseline interface efficiency. Interface failure may also be ameliorated by inserting

redundant links between electric substations and utility demand nodes. Direct incorpora-

tion of generation capacity into the interface and strengthening of substation nodes and

transmission lines may also be applied to reduce the effect of physical fragility and indirect

disconnection of electrical substations on dependent water or gas utility service.

CONCLUSIONS

The developed construction and interface analysis methodology allows for the rapid gen-

eration and performance characterization of previously unexamined interfaces that connect

disparate urban utility systems, especially when subjected to random failures and hurri-

cane events. This interface emphasis converts the formidable task of complex system perfor-

mance modeling into a more tractable problem, albeit approximate, of analyzing significantly

smaller and less intricate subset networks via bipartite graphs and element fragility prop-

erties. Bipartite interfaces may be synthesized from minimal information concerning the

spatial distribution of water or gas nodes within the demand networks and the topological

properties of power supply nodes, allowing for performance assessment of alternative inter-

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connection patterns. In some cases, where the efficiency-based performance metric resembles

real system performance, as in power-water systems, the interface efficiency reduction from

hurricane damage is shown to match actual experiences of Harris County residents. In ad-

dition, the distance and degree based interfaces resemble practical efficiency loss results and

connection configurations that are based on geographical proximity and power distribution

feeder availability, despite not being the most efficient or tolerant to random failure topolo-

gies. This leads to a simple proposed method for generating hybrid interfaces that combines

the topological features of practical and ideal interfaces. Interface efficiency profiles of sim-

ple and hybrid interface networks are compared using varying levels of random failures and

hurricane events having adequate response and feasibility in practice.

Factors including the size of large connected clusters from interface construction, substa-

tion count and associated disconnection potential, substation degree distribution, and the

separation distance between interface elements are found to be the determinants of interface

efficiency. Realistic interface networks minimizing the Euclidean distance between power

supplying and consuming nodes, or maximizing connections to substations with sufficient

feeder availability (i.e., vertex degree) show the least sensitivity to efficiency reductions from

their baseline performance values when subject to random failures and hurricane disruptions.

This is partly from the large number of decentralized substations and lack of significant haz-

ard exposure of transmission lines connecting the systems, although they also display a low

baseline efficiency and need additional infrastructure for enhanced operation and control.

Other interface networks created by applying clustering coefficient and betweenness central-

ity substation rankings produce highly efficient and controllable networks due to high supply

node degree and large connected clusters of elements, but are simultaneously less practical

and sensitive to efficiency losses from random and hurricane induced element failures. The

correlation of these topological findings with actual flow within the utility networks will be

investigated at a future date to test their consistency. Given that efficiency directly measures

the ability of the interface to deliver power to the utility demand nodes, it is expected that

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the flow models combined with hazard simulation will agree with the topological findings

presented here.

Hybrid networks combining both betweenness and distance based construction methods

are shown to address some of the deficiencies of their parent networks, such as improving

distance-based interface efficiency, controllability, and theoretical random failure thresholds,

while decreasing betweenness-based fragility of exposed edges and sensitivity to efficiency

performance losses. In fact, such hybrids are readily implementable, given the high de-

gree of electrical supply node commonality between the parent interface networks. The

electric-water interfaces significantly benefit from this hybrid approach due to their large size

and topological richness, while the electric-gas interfaces are too sparse to produce distinct

topologies that benefit from topological retrofits. Hybrid topological effects are confirmed

by comparing the normalized efficiency profiles of hybrid interfaces to those of the parent

interfaces under hurricane and random disruptions.

This study is a first attempt to analyze the interface topology of interdependent systems

by applying simulated hurricane hazards and random failures. It provides a new tool for

utility operators and emergency management officials to understand approximate trends

in the propagation of failures through the topological interfaces and fragilities of elements

connecting urban infrastructure systems. The impact of other hazards, such as earthquakes,

lightning strikes, and flow-based dynamics on interface networks can also be added in the

future.

ACKNOWLEDGMENTS

The research presented in this paper was supported by the U.S. National Science Founda-

tion through grant CMMI-0728040, and the Shell Center for Sustainability at Rice University.

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List of Algorithms

1 Generation of Interface Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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Algorithm 1 Generation of Interface Networks

Require: VP , Vu {VP : set of transmission substations; Vu: set of water or gas utility demandnodes (UDN)}

Require: dist(si, dj) {Euclidean distance between substation si ∈ VP and UDN dj ∈ Vu}Require: Tk(si) {The topological criterion to select transmission substations si ∈ VP used

for the kth interface connection}Require: Nmax {Maximum number of utility demand nodes connected to each substation

si}Require: ρ {Number of supply nodes si linked to each demand node dj}

1: VIu ← ∅ {Set of electrical substation nodes si connected to UDNs}2: Eu ← ∅ {Set of edges connecting substations and UDNs}3: k ← 1 {Counter variable}4: repeat5: T (si) ← Tk(si) {Tk may be constant (single-metric interfaces) or vary (hybrid inter-

faces)}6: for all dj ∈ Vu do7: OptimalSub← ∅ {Substation maximizing T (si) (see Table 2)}8: RatioValue← 09: for all si ∈ VP do

10: if T (si) > RatioValue and d(si) < Nmax then11: RatioValue← T (si) {Store new optimal substation}12: OptimalSub← si13: end if14: end for15: e← (OptimalSub, dj) {Assign OptimalSub and dj to the new edge e}16: Eu ← e {Store assigned edge in edge set Eu}17: VIu ← OptimalSub {Store assigned substation in VIu}18: end for19: until (k == ρ) {Iterate until all UDNs have ρ connections to substations}20: return [Return interface network GI ] GIu(VIu , Vu, Eu)

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List of Tables1 Composition of the Harris County, Texas, Power Transmission Network . . . 332 Summary of Interface Construction Methods Linking Water or Gas Demand

Nodes to Electrical Substation Supply . . . . . . . . . . . . . . . . . . . . . 343 Summary of the Electric-Water Interface Topological Properties . . . . . . . 354 Substation Supply Node Commonality between Electric-Water Interfaces . . 365 Summary of the Electric-Gas Interface Topological Properties . . . . . . . . 37

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TABLE 1: Composition of the Harris County, Texas, Power Transmission Network

System Order (Average Vertex Degree d(v)) Generators Size Edge Length (Average) kmPower 407 (2.93) 56 556 2 350 (3.95)

Note: generators are power plants located within Harris County that are connected to thepower transmission system. The edge length is the total length of power transmission lines(in km) within the connected network as well. The average length of a line is also given.

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TABLE 2: Summary of Interface Construction Methods Linking Water or Gas DemandNodes to Electrical Substation Supply

Method Basis Connection Criterion T (si)a Purpose

Betweenness maxi

[CB(si)

dist(si,dj)2

]∀j Connects each demand node dj to ρ supply nodes si that

are highly centralized in terms of simplified power flow

Clustering maxi

[C(si)

dist(si,dj)2

]∀j Links each demand node dj to ρ supply nodes si that

are locally well-connected

Degree maxi

[d(si)

dist(si,dj)2

]∀j Pairs each demand node dj with ρ high degree supply

substations si (practical configuration due to availabilityof main power feeders)

Distance maxi [dist(si, dj)−2] ∀j Pairs each demand node dj with ρ closest supply sub-

stations si (practical configuration due to geographicalproximity)

aThese objective functions (T (si), si ∈ VP ) are used to generate all of the interfaces presented in thiswork. A combination of the betweenness and distance topological methods is utilized to produce the hybridinterface networks while all other networks use single construction methods.

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TABLE 3: Summary of the Electric-Water Interface Topological Properties

Method Basis Order LCC Size Substations (d(si)) Edge Length (km)a pc E(GIw)Betweenness 119 69.7% 53 (2.490) 733.499 0.545 0.124Clustering 96 65.0% 37 (3.567) 1 138.105 0.792 0.130

Degree 142 15.5% 76 (1.737) 421.855 0.168 0.035Distance 148 7.4% 82 (1.610) 363.392 0.050 0.028

B-D2 140 12.9% 74 (1.784) 465.390 0.183 0.037B-D3 165 23.6% 99 (2.000) 762.304 0.371 0.058

Notes: LCC: Largest (Interface) Connected Component; pc: critical fraction of removablenodes (Rosas-Casals et al. 2007) as computed in Equation (5); E(GIw): graph theoreticalefficiency of the electric-water interface network. The synthetic electric-water networksmay be grouped into two distinct categories: highly efficient networks with high degreesubstations providing the majority of connectivity (betweenness and clustering coefficient),and sparsely connected, low efficiency networks lacking any notable topological features(degree and distance interfaces). The B-D2 and B-D3 hybrid interfaces have characteristicsintermediate between these two classifications.

aRepresents the total length of edges connecting transmission substations and water pumping stations.

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TABLE 4: Substation Supply Node Commonality between Electric-Water Interfaces

Method Basis Betweenness Clustering Distance DegreeBetweenness 53 11 39 45Clustering 11 37 13 17Distance 39 13 82 63Degree 45 17 63 76

Note: each entry reflects the number of power transmission substations found in theintersecting row and column interface construction methods. For instance, the betweenessand clustering interfaces share 11 of their substations. The total number of substations ineach interface type is on the table diagonal (e.g. the betweeness interface has 53substations and the clustering interface 37 substations).

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TABLE 5: Summary of the Electric-Gas Interface Topological Properties

Method Basis Order LCC Size Substations (d(si)) Edge Length (km)a pc E(GIg)Betweenness 20 25.0% 13 (1.08) 160.928 - 0.161Clustering 16 50.0% 9 (1.56) 202.797 - 0.275

Degree 21 14.3% 14 (1.00) 56.699 - 0.143Distance 21 14.3% 14 (1.00) 48.685 - 0.143

Notes: LCC: Largest (Interface) Connected Component; critical fraction of removable nodes(Rosas-Casals et al. 2007) as computed in Equation (5); E(GIg): graph theoretical efficiencyof the electric-water interface network. The synthetic electric-gas interfaces, unlike theelectric-water interfaces, do not have consistent topological classifications due to thepresence of only seven natural gas compressors in Harris County.

aRepresents the total length of edges connecting transmission substations and gas compression stations.

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List of Figures1 Harris County Complex System: power transmission system with water and

natural gas demand nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Electric-Water Interface Topologies . . . . . . . . . . . . . . . . . . . . . . . 403 Electric-Gas Interface Topologies . . . . . . . . . . . . . . . . . . . . . . . . 414 Electric-Water Interface Network Failure Responses . . . . . . . . . . . . . . 425 Electric-Gas Interface Network Failure Responses . . . . . . . . . . . . . . . 43

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_ _____________ _____ __________ ____ __ _ _____ _ ___ ____ _____________ ___ _____ __ ______ ___ ___ ___ ___ ___ __ ___ _ __ ____ __ __ ____ __ __ ____ __ _____ __ ___ _____ _ __ ____ _ ____ ________ _ ____ _ __ _ __ ___ ____ __ __ ___ ___ __ __ ____ ___ _ ___ _ ___ ___ ____ _ ___ __ ___ _ _ ____ ___ _____ ___ ___ __ ___ _ __ __ ___ __ __ __ ___ ___ __ __ __ ___ ______ _ ___ ______ __ ___ _ __ ____ ___ ___ ____ ___ _ __ ___ __ ______ _____ _ __ _ _____ _ __ __ ________ _ __ _ ___ _ _ ____ _ ___ __ __

_ _ ____ _ __ _____ ____ _____ _____ __ __ ___ __ __ _ __ _ ___ ___

____ __

___!(

!(!(

!(!(!( !(!(!(!(

!(!(!(!( !(

!(!( !(!(!( !(!(!(!(

!(

!(!(!(!(!(

!(!(!(

!( !(!(!(!( !(!( !(!(!(!(!(

!(!(!( !(!(!( !(!(!( !(

!(!(!(

!(

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!(

!(

d

dddddd

ddd dd ddddd dd dd ddddd dddd dd dd ddddd ddddd

ddddddddd

d

d

d

XW

XW

XW

XWXW

XW

XW

0 10 205 km .

LegendXW Gas Compressord Power Plant!( Water Pumping Station_ Electrical Substation

Power Transmission Line

Gulf Coast

FIG. 1: Harris County Complex System: power transmission system with water and naturalgas demand nodes.

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(a) (b)

(c) (d)

(e) (f)

FIG. 2: Electric-water interface topologies under different construction criteria (Gray: WaterPumping Station, Black: Electric Substation): (a) Betweenness, (b) Clustering Coefficient,(c) Degree, (d) Distance, (e) Hybrid: B-D2, (f) Hybrid: B-D3

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(a) (b)

(c) (d)

FIG. 3: Electric-gas interface topologies under different construction criteria (Gray: GasCompressor, Black: Electric Substation): (a) Betweenness, (b) Clustering Coefficient, (c)Degree, (d) Distance

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fraction of Edges Removed, fe

No

rma

lize

d E

(GIw

)

Betweenness

Clustering

Degree

Distance

B-D2

B-D3

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fraction of Nodes Removed, fn

No

rma

lize

d E

(GIw

)

Betweenness

Clustering

Degree

Distance

B-D2

B-D3

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

33 38 43 48 53 58 63

Storm Average Wind Gust Speed (m/s)

No

rma

lize

d E

(GIw

)

Betweenness ClusteringDegree DistanceB-D2 BD-3Ike-Betweenness Ike-ClusteringIke-Degree Ike-DistanceIke-B-D2 Ike-B-D3

(c)

FIG. 4: Electric-Water Interface Network Failure Responses: changes in interface efficiencies,normalized by their respective undamaged efficiency scores, as a function of hazard inten-sity. (a) Electric-Water Interfaces under Random Edge Attack, (b) Electric-Water Interfacesunder Random Node Attack, (c) Electric-Water Interfaces under Hurricane Hazards

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fraction of Edges Removed, fe

No

rma

lize

d E

(GIg

)

Betweenness

Clustering

Degree

Distance

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Fraction of Nodes Removed, fn

No

rma

lize

d E

(GIg

)

Betweenness

Clustering

Degree

Distance

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

33 38 43 48 53 58 63

Storm Average Wind Gust Speed (m/s)

No

rma

lize

d E

(GIg

)

Betweenness ClusteringDegree DistanceIke-Betweenness Ike-ClusteringIke-Degree Ike-Distance

(c)

FIG. 5: Electric-Gas Interface Network Failure Responses: changes in interface efficiencies,normalized by their respective undamaged efficiency scores, as a function of hazard intensity.(a) Electric-Gas Interfaces under Random Edge Attack, (b) Electric-Gas Interfaces underRandom Node Attack, (c) Electric-Gas Interfaces under Hurricane Hazards

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List of Figure Captions for “Interface Network Models for Complex Urban Infrastructure

Systems “ Manuscript, Research paper number ISENG‐134

Figure 1. Harris County Complex System: power transmission system with water and natural gas

demand nodes.

Figure 2. Electric-water interface topologies under different construction criteria (Gray: Water

Pumping Station, Black: Electric Substation): (a) Betweenness, (b) Clustering Coefficient, (c)

Degree, (d) Distance, (e) Hybrid: B-D2, (f) Hybrid: B-D3

Figure 3. Electric-gas interface topologies under different construction criteria (Gray: Gas

Compressor, Black: Electric Substation): (a) Betweenness, (b) Clustering Coefficient, (c) Degree,

(d) Distance

Figure 4. Electric-Water Interface Network Failure Responses: changes in interface efficiencies,

normalized by their respective undamaged efficiency scores, as a function of hazard intensity.

(a) Electric-Water Interfaces under Random Edge Attack, (b) Electric-Water Interfaces under

Random Node Attack, (c) Electric-Water Interfaces under Hurricane Hazards

Figure 5. Electric-Gas Interface Network Failure Responses: changes in interface efficiencies,

normalized by their respective undamaged efficiency scores, as a function of hazard intensity.

(a) Electric-Gas Interfaces under Random Edge Attack, (b) Electric-Gas Interfaces under Random

Node Attack, (c) Electric-Gas Interfaces under Hurricane Hazards

Figure Captions List



interface network models for complex urban infrastructure systems

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