Review on Energy Resilience

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Transportation Research Part E

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

Climate-adaptive planning for the long-term resilience of transportation energy infrastructure

Arash Beheshtiana,b, Kieran P. Donaghya, R. Richard Geddesb, H. Oliver Gaob,c,⁎

a Department of City and Regional Planning, Cornell University, Ithaca, USA b Cornell Program in Infrastructure Policy, Department of Policy Analysis and Management, Cornell University, Ithaca, USA c School of Civil and Environmental Engineering, Cornell University, Ithaca, USA

A R T I C L E I N F O

Keywords: Transportation energy Climate-adaptive planning Resilience Sea level rise Flooding

A B S T R A C T

This paper investigates a long-term planning response to the climate-vulnerability of transpor- tation energy infrastructure in the borough of Manhattan, NY. The proposed model, a two-stage stochastic optimization, features a hybrid utility-regret function with increasing relative and decreasing absolute risk aversion. Modeling results suggest (1) investment in early- and late-stage resilience-enhancing solutions as a complementary approach with significant weight on im- mediate actions, and (2) a decentralized supply chain formation through an early-stage de- ployment of reservoir tanks within the case study area.

1. Introduction

Repeatedly over the past decade, storms have revealed the vulnerability of the urban built environment to storm-surge flooding along America’s seaboard. Among the built environment components, interdependent infrastructures are recognized as vulnerable systems with potential to cause a “debilitating effect on security, national economic security, national public health or safety, or any combination thereof” (United States Department of Homeland Security). Such criticality relates in part to system exposure and resilience in the face of exogenous shocks, and also in part to the type and magnitude of failure imposed by threats.

The motor fuel supply chain (MFSC), aka the transportation energy infrastructure1, is well-recognized as critical (Energy Sector- Specific Plan 2015, Department of Homeland Security), yet also as highly vulnerable to climatic extremes (National Institute of Standards and Technology). The climate-vulnerable infrastructure of transportation energy may eliminate the operability of trans- portation system in time of disaster, rippling the failure to pre- (e.g. evacuation and sheltering) and post-event emergency tasks, and hampering the recovery processes (Beheshtian, 2016).

Just within the past five years, two natural disasters (i.e. super-storm Sandy and Hurricane Harvey) have further underscored the vulnerability of the MFSC, and have attracted interest among scholars, practitioners, and policy-makers in revisiting adaptability criteria and resilience standards. By the time it made landfall in New York City (NYC), Sandy was weakened to a post-tropical cyclone (National Hurricane Center), however, its impact on infrastructure, and specifically on the MFSC, was devastating. Even ten days after the storm passed NYC, more than 28% of the city’s gas stations had no gasoline to operate, according to the U.S. Energy

https://doi.org/10.1016/j.tre.2018.02.009 Received 11 August 2017; Received in revised form 23 January 2018; Accepted 21 February 2018

⁎ Corresponding author at: School of Civil and Environmental Engineering, Cornell University, Ithaca, USA. E-mail address: [email protected] (H. Oliver Gao).

1 The MFSC refers to the multi-commodity supply chain of gasoline and diesel. The entire chain is divided into up-, mid- and down-stream. Covering far-flung petroleum producing area, finding, lifting, and processing oil and gas from subsurface into surface are considered to be the up-stream’s activities. Transportation and storage of crude oil from up-stream plants for further processing by pipeline, railway, road, or tanker are activities in the mid-stream. The down-stream’s activities, the focus of this research, include further processing of crude oil and natural gas into the final product, and marketing, delivering, and retailing of the fuel in service stations.

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Information Administration (EIA). Sandy shut down twenty-eight terminals across the region (EIA’s Petroleum Terminal Survey), flooded thousands of distribution roads, and submerged many service stations, which together resulted in: limited mobility, cascading failures to dependent critical infrastructures such as transportation systems (Comes and Van de Walle, 2014), and eventually a recovery effort that was slowed down and further complicated. The Hurricane Sandy Rebuilding Task Force estimated that Sandy imposed $30-$50 billion of economic loss (aside from physical damage) due to extensive power outages, liquid fuel shortages, and near-total shutdown of the region’s transportation system.

More recently, Hurricane Harvey, the wettest hurricane in the history of the contiguous United States (National Oceanic and Atmospheric Administration) and the only major hurricane to make landfall in the United States since Wilma (2005), severely damaged more than a dozen refineries in the area, and it shut down many facilities, including Motiva, the nation’s largest oil refinery. The refinery outages, the closure of several key-ports alongside, damaged gas stations, and flooded roads caused a major fuel shortage. It took the fueling infrastructure weeks to cope with the situation and fully bounce back into a state of business-as-usual. During Hurricane Harvey, the MFSC’s failure had a ripple effect on the transportation system, and it inevitably hampered each and every activity conditioned on the functional mobility system, including pre-disaster evacuation process planning, intra-disaster emergency tasks, as well as plans for post-disaster recovery efforts.

Governments have responded to the MFSCs climate-vulnerability at different levels to enhance resilience. Federal agencies have been involved mainly by “supporting” private sector efforts to harden the resilience of the energy system. The U.S. Government Accountability Office classified the federal government’s role to provide such influence on the private sector – providing information, regulatory oversight, technology research and development, and market incentives and disincentives – as “limited.” Asides from the federal government, different states have advanced mechanisms to integrate adaptive measures into their fueling infrastructure. Following super-storm Sandy, New York State (NYS) launched a number of initiatives and passed several regulations to support state- wide hardening and resilience-enhancing strategies for the energy sector. NYS funded two structural strategies: (1) establishing a transfer switch for backup power generation in every downstate gas station within a half mile of a highway exit or hurricane evacuation route, along with onsite backup generators in gas stations in “strategic locations” to provide power when the utility is not available, as subsidized by the New York State Energy Research and Development Authority (NYSERDA) and (2) as part of a pilot program on Long Island, a fuel reservoir tank with 3 million gallons of fuel to supply gasoline during a fuel shortage or prolonged disruption to MFSC, as proposed through a $10 million project by the state governor (NYS Governor 2013).

Many strategies and programs have also been pursued by Texas and Florida following the devastating 2005 hurricane season (Hurricane Katrina in Florida and Hurricane Rita in Texas) and the 2008 hurricane season (hurricanes Ike and Gustav in Texas). Florida, for example, has required service stations within a half mile of evacuation routes to install a backup generator, and Texas created the Fuel Team, “a private-sector partner to the State,” to serve as an information clearinghouse and critical communications hub (Hoffman et al., 2009; Hoffman and Bryan, 2013).

The private sector, which controls a majority of MFSC elements, has also invested in hardening and resilience measures. Refiners have built floodwalls along the Houston Ship Channel and around Pascagoula to contain a 100-year storm surge, and have designed elevated substations, control rooms, and pump stations above the likely flood level. The Colonial Pipeline, the largest refined pro- ducts pipeline system in the U.S., also purchased 12 trailer-mounted portable generators, seven transformers, and miles of associated cabling in 2006. For further details on private sector activities in this area, the reader may consult the, Department of Energy’s (DOE’s) Report on the Energy Industry Response to Recent Hurricane Seasons (Hoffman, 2010).

Responses to MFSC vulnerability have developed in two separate directions. First, these responses address empirical shortcomings reported by or surveyed through system operators and experts. For instance, the DOE’s Office of Electricity Delivery and Energy Reliability (2010) phone-interviewed 14 energy companies to investigate measures for hardening assets and making the energy supply more resilient (Hoffman, 2010). The NYSERDA interviewed terminal operations in NYS to survey terminal vulnerabilities made visible from impacts by Hurricane Sandy, Hurricanes Irene, and tropical-storm Lee. The second type of response is one based on generic strategies commonly taken to enhance a system’s ‘flexibility’ and ‘redundancy’ (Sheffi and Rice, 2005; Falasca et al., 2008). For instance, the generic properties of a resilient supply chain such as out/multi-sourcing, easy modification of inventory levels, and cross-trained workers have been implemented by different key players in the transportation energy infrastructure.

While these strategies, to some degree, have improved the MFSC’s resilience, their effectiveness has been limited, as experienced following the 2017 hurricane season in Houston, TX, and Tampa, FL. Such shortcomings include (1) approaches are too case-specific, in other words, where the MFSC’s elements are treated as isolated components operating with no functional or physical inter- dependence upon other system components, (2) a process where decision-making and physical investment take place in an isolated fashion, and (3) static climate-adaptive strategies that are based only on challenges the MFSCs face at present, despite the dynamic nature of climate change and society.

Advancing fully-integrated strategic planning for a resilient MFSC is currently well out of reach due to several contributing factors. The multi-stakeholder environment of the transportation energy infrastructure obstructs mechanisms for information-sharing or for the creation of institutional equilibria for decisionmakers. The complexity inherent in long-term investment in any asset- intensive infrastructure also contributes to a lack of resilient strategic planning. Despite this, efforts toward one integrated modeling platform can provide decisionmakers, whether from the private or public sector, with a comprehensive grasp of (i) the MFSC’s multi- layered structure and inherent interdependencies within and across infrastructure components, (ii) its long-term vulnerability to climatic extreme trajectories, and (iii) the dynamics between a system’s overall resilience and an optimum portfolio investment. As already discussed, such one approach may not be able to develop an ultimate planning response to the climate-vulnerable MFSCs. Yet an integrated approach aims to provide stakeholders with diverse perspectives and a common ground to facilitate their cross-jur- isdictional and multidisciplinary dialogue, and possibly eliminate excess or redundant investment.

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In this article, we study the resilience of a MFSC infrastructure by focusing on the borough of Manhattan, NY. The case study is distinct because (1) Manhattan possesses a particularly complex geography in which access to and regress from the borough are conditioned on the availability of only twelve entry points, which are either tunnels or bridges, all sited in flood-prone areas, (2) NYC’s MFSC is spatially dispersed along low-lying coastal areas, offshore, or in areas exposed to extreme winds, (3) a home to the largest global financial market, NYC’s climate-vulnerability, as witnessed following super-storm Sandy, may impose billions of dollars of economic loss, with ripple effects throughout the country.

To the best of author’s knowledge, the proposed framework is the only analytical platform that models the long-term resilience of transportation energy infrastructures down-stream, and that projects the long-term climate change-induced hazards on the MFSC. Our formulation features a nonlinear stochastic mathematical program in which two-stage decision variables are simultaneously optimized against the MFSC’s maximum resilience when stressed or under attack. Further, we re-conceptualize the term “resilience” and apply in-depth analytical-resilience planning in a true-to-scale case study.

2. Reviewing models and related works

We next summarize the key-literature in the resilience domain, and we exclusively review works on transportation energy in- frastructure. Then we review facility location problems that underlie the proposed model and discuss the shortcomings of existing models.

2.1. Resilience planning

Resilience planning investigates a strategic response to enhance a system’s ability to withstand, adapt to, and recover from the aftermath of hazards in a timely and efficient manner (Turnquist and Vugrin, 2013). This definition is close to what supply risk management commonly refers to as resilience, i.e., where resilience as a system property is recognized as “flexibility” and “re- dundancy” (Sheffi and Rice, 2005). In addition to qualitative research (Godschalk, 2003; O'Rourke, 2007; Linkov et al., 2014; Tanner et al., 2014) that mainly has discussed the conceptual framework of system resilience (Hosseini et al., 2016), quantitative studies have developed analytical approaches to address a variety of related questions and within different fields of study.

The literature describes supply chain (SC) resilience with a focus on (1) uncertainty and forecasting with respect to supply/ demand, (2) SC network design and the resilience of critical elements in times of disaster, and (3) tactical/operational aspects of the SC (e.g. inventory management and process control). Quantitative research in SC resilience has adopted a variety of approaches to study the infrastructure topology and its vulnerability/resilience to external shocks, including fuzzy set theory (Aleksić et al., 2013; Azadeh et al., 2014), game theory (Bakshi and Kleindorfer, 2009), analytical hierarchy (Rajesh and Ravi, 2015; Thanki and Govindan, 2016), decision tree modeling (Alexander et al., 2014), simulation (Carvalho et al., 2012), and optimization techniques. Due to the framework of the proposed model, we will review the literature utilizing optimization approaches. The readers may consult Fahimnia (2015) for extensive review of the other techniques.

Sahebjamnia et al. (2015) developed a multi-objective mixed-integer linear programming model to formulate resource allocation for both resuming and recovery plans simultaneously. Minimizing the total loss in operating level and recovery time of key products, their model sought the final preferred compromise solution. Pishvaee et al. (2012) advanced a bi-objective optimization (i.e. minimizing the total cost and maximizing the SC responsibility) to integrate the social responsibility of corporate and business units into SC network design. Santoso et al. (2005) applied a stochastic programming model and solution algorithm for two large-scale supply chain network design problems under uncertainty to identify and statistically test a variety of candidate design solutions.

Turnquist and Vugrin (2013) have modeled infrastructure resilience employing absorptive-, adaptive-, and restorative-enhancing strategies. Utilizing a two-stage stochastic model, they optimize a given distribution network’s post-disruption operability with respect to pre-event investment decisions. Miller-Hooks et al. (2012) have measured a network’s maximum resilience level by staging variables into an optimal set of “preparedness” and “recovery” actions. Modeling the resilience of an intermodal system, their model maximizes the number of post-disaster shipments while spending optimally on pre-event preparedness. Torabi et al. (2015) have also explored the impact of disruption risks on supply-chain networks. Through a bi-objective, two-stage probabilistic optimization model, they investigated optimal supplier selection through an order allocation problem.

2.2. Resilience of the transportation energy infrastructure

A majority of the models developed on emergency (aka relief) SC management focus on generic, non-fuel emergency supplies (e.g. food, bottled water, and blankets) which, in nature, are different from petroleum products (Rawls and Turnquist (2010) and Mete and Zabinsky (2010)). Few works, nevertheless, investigate the resilience of transportation energy infrastructure, mainly at up-stream and mid-stream scales.

Costaa et al. (2017) studied the impact of earthquakes on fuel storage tanks of the coastal communities. They collected in- formation from interviews with the key stakeholders in British Columbia and used simulation software to model the impact. Sandia National Laboratories also advanced a methodology to estimate the impact of different types of disruption and system configurations on infrastructure overall wellbeing (Wilson et al., 2015). Their study was based on seven stressing events of a Gulf Coast hurricane, a Mid-Atlantic hurricane, a Boston Harbor oil spill, a Denver refinery explosion, a New Madrid earthquake, a Southern California earthquake, and a Northern California earthquake. Using the National Transportation Fuels Model to simulate the system-level response of the liquid fuels sector, they found wide variations between liquid fuel system responses to different extreme events. They

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concluded those infrastructures which are widely dispersed (i.e. those of the Gulf Coast and New Madrid and Southern California infrastructures) are the most vulnerable.

Cruz and Krausmann (2013) have reviewed reports on climatic extremes that have hampered the activities of the oil and gas industry. Their work included the four extremes of warmer temperature, storms and floods, tropical cyclone, and lightning on three sectors: up-, mid-, and down-stream. Torres and Alsharif (2016), through a newspaper content analysis method, documented the fuel shortage in Broward County, FL, following Hurricane Wilma (2005). They found power outages to be a key-vulnerability of the MFSC and suggested the implementation of tax rebate or loan programs to subsidize backup generators. Li et al. (2017) modeled the supply of gasoline in New Jersey and after super-storm Sandy. Through a mixed integer, multi-objective program, they located those gas stations that are the best candidates for backup generator installation, and they modeled a delivery scheme that accounts for in- creased demand with respect to the lack of public transportation, and accounts for other considerations such as equity. Suzuki (2012) compared the damages induced by the shortage of fuel and the equivalent-sized shortage in emergency, and studied the optimum type of vehicles to be used in time of disaster. Through an optimization problem, he concluded the shortage of fuel to be more damaging than the shortage of emergency supplies, and found smaller trucks may be preferred to larger trucks when the shortage of fuel becomes severe.

In a series of work, Beheshtian and colleagues studied climate-vulnerability in NYC’s transportation energy sector. They studied the system’s overall vulnerability in the face of 100- and 500-year flooding events (Beheshtian, 2016), resilience-enhancing strategies which are optimum against the random arrival of hurricanes (Beheshtian et al., 2017a), and the application of queuing theory in NYC’s service stations (Beheshtian et al., 2018).

2.3. Facility location problem

Acquiring optimal siting and sizing of a finite number of spatially-dispersed facilities is the main concern in SC network design, and hence is a key modeling concern in the literature studying the adaptation planning and resilience of supply chain infrastructures. This has been modeled using a range of mathematical programing models as summarized in the following.

a. Fixed-charge facility location: investigates the subset of candidate-locations that minimize the distribution cost of flow from supply nodes to demand locations (Snyder, 2006).

b. Maximal-covering location problem: Serves those analytical methods optimize against constrained and insufficient resource/ budget. It assumes a fixed number of facilities and investigates the optimal location of facilities to cover the maximum demand (Santoso et al., 2005).

c. Set-covering location problems: minimizes the facility location costs to satisfy a specified level of coverage. d. P-median: locates p number of facilities in order to minimize the distribution cost for satisfying demand, while each of the demand

points is supplied from the nearest located facilities (Bertsimas et al., 2011). e. P-dispersion: locates p facilities on a network so that the minimum distance between any pair of facilities is maximized (Mete and

Zabinsky, 2010). f. α-reliable, p-center problem: minimizes the number of facilities needed to meet at least α percent of total demand (Barbarosoǧlu

and Arde, 2004). g. Maximum-covering, shortest path problem: identifies the shortest path between supply and demand nodes with respect to

maximum satisfied demand (Current et al., 1985).

Aside from the above classification, facility location problems are also categorized by the nature of input parameters and decision variables. Accordingly, these problem sets can be divided into deterministic and nondeterministic models. In deterministic models, all inputs are known quantities. Such models address the problem’s static or dynamic nature. Through the static-certainty environment, the models implement a single solution at one point in time. Alternatively, dynamic-certainty models capture real-world changes over time and seek the optimal solution through an extended planning horizon. Although both the required integer programing for- mulations and the NP-hardness constraining the facility location problems into deterministic forms create high computational costs, improvements in operations research techniques allow practitioners to consider the uncertain nature of supply-chain infrastructures (Owen and Daskin, 1998).

Whereas, the inclusion of randomness introduces nondeterministic, aka probabilistic, modeling approaches involving parameters whose values are governed by probability distributions (Snyder, 2006). If the probability distribution, whether discrete or con- tinuous, is known, mathematical programing takes the form of a stochastic optimization (SO). Alternatively, the model features a robust optimization where the probability distributions of parameters are unknown and assumed to be continuous (Santoso et al., 2005).

The robust optimization form of the facility location problems falls into one of the following two minimax structures: minimizing the inoperability sequenced through the worst-case scenario(s), and minimizing the maximum opportunity loss, aka “regret”, across all scenarios. Since the proposed model is not centered explicitly on robust optimization, the reader is referred to Snyder (2006). Conversely, modeling the facility location problems through SO, where randomness properties are known and assumed to be discrete, is scenario based. This means that all possibilities regarding exogenous event characteristics, network conditions, and end-user behavior are clustered within a finite number of discrete scenarios. Each scenario represents an event with a particular probability of occurrence. The SO thus seeks a set of decision variables that are optimal given a probability-weighted average of the objective function of scenarios.

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Mete and Zabinsky’s (2010) study examined mixed-integer SO to select the storage locations of medical supplies, required in- ventory levels, and distribution assignment for each type of medical supply. Their model considers uncertainties associated with six earthquake scenarios (i.e. two faults within three time-frames) and minimizes the total transportation time of assigned vehicles. Barbarosoǧlu and Arda (2004) model post-earthquake optimal multi-modal, multi-commodity movement of material over an urban transportation network. They analyzed nine scenarios in total. Scenarios are based on a careful consideration and combination of uncertainties in physical (i.e. transportation-arc capacity) as well as social (i.e. demand) parameters.

Beheshtian et al. (2016) developed a nonlinear SO and have investigated prepositioning resources to enhance fuel-supply-chain resilience. Their model is optimized subject to the arrival of two extreme events: a 100- and a 500-year flood. Rawls and Turnquist (2010, 2011) included varying demands and network damages in a study of 51 scenarios. They considered both single storms and combinations of two “almost simultaneous” storms. Through an α-reliable model, they optimized the expected cost of post-event shipment by pre-positioning resources in advance of hurricanes. Those scenario-based SO models are entirely optimized against a full range of predefined scenarios. Daskin et al. (1997), however, introduced a class of SO called ‘α-reliable minimax regret’ in seeking a solution minimizing the maximum regret over a ‘reliability set’ (i.e. “an endogenously determined subset of the scenarios”). They subsequently proposed an α-reliable mean-excess regret model in which the expected regret is minimized across an endogenously- selected subset of worst-case scenarios whose collective probability of occurrence is no more than (1-α)100%.

2.4. Research gap and position of this research

We recognize two major drawbacks in existing approaches toward disaster preparedness planning, and discuss how the proposed model addresses these shortcomings. First, fuel shortage is a product of the system failure and happens on a regional scale where a number of public and private agents holds assets and responsibilities. Such multi-stakeholder governance in the absence of system thinking causes sub-optimal, excess, or redundant investments in which the MFSC’s elements are treated solely and independently from their inherent functional interdependencies. At present, there exists no analytical platform, neither in the academic literature nor in practice, to model the MFSC down-stream as an integrated system of systems, and to simulate motor fuel distribution on the regional scale.

Second, investment decisions on an asset-intensive infrastructure like the MFSC must be justified in the context of the changing patterns/intensity/frequency of climate-change-induced hazards. This begs a dynamic analytical platform facilitating the analysis of costs (i.e. risks associated with lacking or late investments) and benefits (i.e. enhanced-resilience) that correspond to decisions taking place in a continuous, long-term horizon. Embodying this problem through conventional dynamic cost-benefit analysis, however, carries two drawbacks: (1) conventional models are utility-based, which are developed to minimize the system’s overall disutility (Kirshen et al., 2015; Rose, 2016), while disaster management practitioners favor regret-based objective functions2

(Intergovernmental Panel on Climate Change, 2014) in which the policy maker seeks minimum regret through those adaptation strategies that “perform fairly well no matter which scenario is realized” (Liberatore and Scaparra, 2011; Klijn et al., 2016; Wang et al., 2017); (2) The second shortcoming relates to the fundamental difference in the nature of risk in business and humanitarian fields. While the former sector performs minimax or α-reliable models, leading to excessively risk-averse tendencies or risk-neutral decisions, respectively, humanitarian fields focus on a “hybrid” approach well-utilizing the assets available through built environ- ment criteria and risk typology (Chorus et al., 2014).

This research thus proposes a dynamic analytical platform to address the shortcomings elaborated above. This model is an extension to the α-reliable, mean-excess regret model, which is set to endogenously select a reliability set of scenarios, and allows the decisionmaker to identify a portfolio of optimal risk-averse decisions, while excluding decisions that over-utilize resources. The remainder of the paper is organized as follows. In Section 3, we introduce the toy model, discuss its attributes (i.e. inoperability, stochasticity, regret theory, location theory, and economics of adaption), and apply it on a testbed of networked infrastructure. Then, in Section 4, we present the case study area, discuss assumptions embedded in the model, and expand the model generalized in Section 3 into a detailed formulation. Numerical analysis and implications are provided in Section 5, followed by Section 6, which concludes the research project by summarizing modeling results and suggesting directions for future research.

3. Methodology

The proposed model is sited at the intersection of five analytical techniques. Therefore, we divide this section into five sub- sections, each introduces and fits an analytical technique into the proposed model. We first introduce the method we used for simulating extreme-induced inoperability across the infrastructure’s elements. Then we explain how a two-stage optimization is utilized to enhance the infrastructure resilience. Section 3.3 introduces the regret theory to replace the conventional inoperability index. Section 3.4 discusses the facility location theory and the last sub-section adds an econometric piece to the model.

2 The concept of regret theory within the context of SC must be separated from no-regret (i.e. reducing climate change vulnerability but provide sufficient other benefits to be justifiable, even in the absence of anticipated climate change impacts (Huq et al., 2014) and low-regret (i.e. relatively small short-term investments for relatively large anticipated climate adaptation benefits (Martin, 2015)) approaches predominating in planning communities (Butler et al., 2016).

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3.1. Developing a testbed: Modeling infrastructure inoperability

We consider a given metropolitan area that is vulnerable to an extreme-flooding event. The arrival of such one extreme event in year t corresponds to an occurrence chance, an expected intensity, and an area prone to flooding. The projected climate change for year t + n, however, induces two disruption types: permanent flooding due to sea-level-rise (SLR) and storm-surge flooding. The latter event is expected to have a shorter return period and higher intensity, and also covers a larger geographically-dispersed area in comparison with the year t flooding event. Fig. 1 shows a hypothetical networked infrastructure and its vulnerability to flooding events.

We initially assume the MFSC’s elements sited in any of the vulnerable land parcels become fully inoperable if the corresponding flooding event occurs. This leads us to four sets of network topology, each corresponding to the scenario s = 1, 2, …, S, in year t = 1, 2, …, T. The infrastructure is expected to experience some degree of inoperability under each flooding scenario (inoperabilities such as non(sub)-optimal flow distribution assignment due to flooded arcs, unmet fuel demand caused by lack of connectedness between supply-demand pairs, etc.).

3.2. Solving for maximum resilience: A two-stage stochastic framework

Investing in resilience-enhancing strategies (RESs) prior to climatic extremes protects the system’s overall functionality in time of disaster. Consequently, one decisionmaker faces two lines of questions: how to allocate/preposition resources prior to the hazard, and how to operate the network optimally in time of disaster. The separation of pre- and post-event decision variables leads to a two-stage framework: allocating resources according to various RESs (the first stage investment decisions) to maximize the infrastructure resilience when stressed (the second stage network operation decisions). In fact, second-stage decision variables (i.e. operational decisions in time of disaster) are conditioned on decisions made in the first stage. The two-stage optimization problem could be framed as the following.

Fig. 1. Hypothetical networked infrastructure in the face of extreme events on years t and t + n.

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̂VMax t s, (1) ̂ =V U Cs.t. f ( , )t s t s t s, , , (2)

=U IVOph ( )t s e t s, , (3)

=C IVOpg ( )t s e t s, , (4)

= +IVOp O Ie t s

e t s

e t s, , ,

(5)

+ ≤O I 1e t s

e t s, , (6)

The objective function (1) represents the maximum operability level the system may possess during a given scenario s in time t ( ̂V t s, ). The scenario-specific maximum operability is driven by flow distribution cost, Ct s, , and unmet demand rate, U t s, , as shown in constraint (2). These two functions, as shown in constraints (3) and (4), are conditioned on the operability of the infrastructure elements e = 1, 2, …, E during scenario s, IVOp .E

t s, Meaning ̂V t s, = Ƒ (IVOpEt s, ). Constraint (5) restricts the operability of element e, during scenario s in time t, on either of the following conditions: element e is not sited in an area vulnerable to a corresponding hazard (where Oe

t,s is a binary input parameter: 0 if the element e is sited in a vulnerable area, 1 otherwise) or if sited, investment according to a RES must be made in place (where Ie

t,s is a binary decision variable: 1 if the RES is implemented to protect the element e against the aftermath associated with a hazard expected in year t, scenario s, 0 otherwise).

Constraint (6) blocks investment in non-vulnerable elements located outside floodplains. If a given element e is not sited in a flood-prone area assigned to scenario s (i.e. =O 1e

t s, ), there is no need to implement RES to enhance its robustness (i.e. =I 0e t s, ).

Otherwise, the model decides whether investment in element e’s resilience is necessary. Therefore, the infrastructure’s system in- operability (to be minimized in the second stage) during each and every scenario is conditioned on its elements’ inherent- or en- hanced-resilience (to be determined in the first stage).

Available resources/assets could be deployed through S different forms, each representing a set of investment decisions which is optimum against the system maximum resilience during scenario s. To investigate a set of investment decisions which is optimum against all scenarios S, however, we weight each scenario’s maximum operability ( ̂V t s, ) by the scenario’s associated chance of arrival (Pt s, , denoted as the scenario s’s occurrence chance in time t). We then utilize a weighted-sum function representing all scenarios. The solution which is optimized against the new objective function, may be non- or sub-optimal under each scenario s, yet, is an optimum against the full range of scenarios S. The new objective function is shown in expression (7). The variable V t s, represents resilience level the system will hold during scenario s.

∑ P VMax ( ) t s

t s t s

,

, ,

(7)

3.3. Modeling infrastructure (in)operability under regret theory

The value of V t s, is nonnegative and upper-bounded by ̂V t s, . The difference between ̂V t s, and V t s, , Rt s, , is called “regret.” The regret is defined as the planner’s understanding of the marginal utility that the best policy may generate compared with the utility under the policy actually chosen. Being solved exogenously, ̂V t s, are fed into the model as known-input parameters. The model thus seeks to minimize regret, which is analogous to minimizing the difference between V t s, and ̂V .t s, The regret model is formulated in the following. The objective function (8) minimizes the system-wide regret which is defined in expression (9).

∑ P RMin ( ) t s

t s t s

,

, ,

(8)

̂= − ∀R V V s ts.t. ,t s t s t s, , , (9)

3.4. Modeling facility location

Two of the key modeling approaches in scenario-based, risk-mitigation planning are α-reliable and minimax models. The α- reliable model, a risk-neutral approach, ignores high-impact, but low-probability scenarios. By contrast, the latter approach, the minimax model, is overly sensitive to high-impact/low-probability scenarios and may result in excessively risk-averse tendencies by over-utilizing resources in a way that may never be used in practice. To address this shortcoming, the proposed model sits at the junction of these two models. We propose a new method to capture and minimize the expected regret corresponding to those scenarios with relatively high-impact and considerable cumulative chance of occurrence. This represents an extension to the α- reliable, mean-excess regret model introduced by Chen et al. (2006). Our model divides the sub-scenarios into an α quantile, β quantile, and (1-α-β) quantile, so that the least regret among the β quantile scenarios, φ', is larger than the maximum regret cor- responding to α quantile sub-scenarios, φ, and smaller than the regrets related to the rest of the sub-scenarios (with cumulative probability of (1-α-β)100%).

By minimizing the expected regrets of the β quantile scenarios, the model simultaneously selects three sets of scenarios with cumulative probability of α100%, β100%, and (1-α-β)100% and minimizes the regret values of (some or all) scenarios with respect to

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their corresponding weighted average. The set of scenarios clustered under β (or any other) quantile and initially selected by the model may not be the same set when the model converges. The proposed regret model is then:

∑ P R ZMin ( ) t s

t s β t s

β t s

,

, , ,

(10)

∑ ≥P Z αs.t. s

t s α t s, ,

(11)

∑ ≥P Z β s

t s β t s, ,

(12)

≤R Z R Zα t s

α t s

β t s

β t s, , , ,

(13)

We define Zα t s, and Zβ

t s, as binary decision variables that indicate whether a given scenario s is included in sub-sets of α and β, respectively. Expression (10) represents the objective function minimizing the probability-weighted regret of the β quantile’s sce- narios. Constraint (11) assures the scenarios within α quantile cover at least 100α% of the events. Constraint (12) assures that at least 100β% of the scenarios are covered in β quantile. Constraint (13) guarantees that the regrets of β quantile’s scenarios are not smaller than the regrets of the α quantile scenarios. In Fig. 2, the proposed model is compared with other approaches commonly-used in the risk management literature. The bars represent the probability distribution function of the regret values across all scenarios. The red bars relate to the regret value of those scenarios subject to minimization. The height of each bar indicates the number of scenarios sharing the same regret value.

Fig. 2 (right) shows the separation of α and β quantile scenarios in the proposed model. The top-left figure shows the generic, average regret model, and the top-middle figure shows the worst-case (minimax) regret model. The down-left figure represents the α- reliable worst-case regret model, and the down-middle figure schematizes the α-reliable mean-excess regret model.

3.5. The economics of resilience

Planning for long-term climate-adaptation requires a finance mechanism to fund the project’s expected capital cost. One such mechanism takes into account the true interest cost (TIC) or interest rate, as well as the non-residential construction index (CI). The TIC reflects the fixed-rate interest expense to the municipal bond issuer or interest rate for corporate bonds/loans. The CI forecasts the inflation/deflation rate for non-residential construction costs.

Herein, we assume the CI is constant in time and 6% below the TIC (assuming CI to be equal larger the TIC simplifies the problem set in which results favor the early investment, inevitably). This introduces a situation in which the later investment will provide the decisionmaker with extended budget cap, yet higher risk due to vulnerable infrastructures. A tradeoff thus exists between moderately- vulnerable elements existing for a relatively longer period of time versus resilience-enhanced elements for a limited time-period.

Let mr be the marginal interest rate (i.e. = −mr TIC CI, assumed to be 0.06), Ct be the investment cost in year t, and the present value of the budget cap BC. The following constraint converts the cost of investments in resilient infrastructure into the present (year t 0) dollar value, and restricts that below the maximum budget:

∑ +

≤ −

C mr

BC (1 )t

t t t0 (14)

Fig. 2. Illustration of the regret models.

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4. Illustrative case study and model formulation

By the 2080s, the sea level in NYC is projected to rise 18–39 in., inundating thousands of acres of low-lying areas (Horton et al., 2015). This trajectory of SLR will increase the frequency and intensity of hydro-meteorological hazards. In the Battery section of Manhattan, for instance, by the 2080s, the annual chance of today’s 100-year flood will be 4 times higher and the corresponding surge flood height may reach up to 13 ft, according to the New York City Panel on Climate-change (NPCC).

To model the impact of climate-induced events on the MFSC’s long-term resilience, we adopt NYC’s most recent coastal flood risk maps/projections from the NPCC 2015 report. The report includes the latest Federal Emergency Management Agency (FEMA) 1% and 0.2% (i.e. “100-year” and “500-year”) floodplain and sea level rise for the 2020s, 2050s, 2080s and 2100. Accordingly, we define a series of 6 time-variant scenarios. These scenarios reflect two extreme events (i.e. 100- and 500-year flooding) in each and every time windows of 2020s, 2050s, and 2080s. To capture varying possibilities in network damages and demand fluctuations within each of the scenarios, we pre-define randomness in the following five input parameters.

(1) The availability of supply nodes (i.e. terminals and refineries operability) in time of disaster is not a deterministic input parameter. According to the New York State Petroleum Terminal Resiliency Assessment report, the vulnerability of the down-state terminals to climatic extremes (extremes studied in this report are superstorm Sandy, Hurricane Irene, and tropical storm Lee) is “site- specific” and occurs in “multiple ways.” To reflect such, we assume the terminals service rate (SR) drops in times of disaster by 50%, and if it drops, the SR decreases by the severity of the flooding events.

(2) The SR of gas stations, as experienced following recent NYS storms, may drop in times of flooding. Here, we assume the station SR may drop after 100-year events. However, we assume they become fully dysfunctional in times of 500-year events, meaning their SR drops to 0.

(3) In Manhattan, while the real estate market has prospered, only 29 gas stations remain, down from 39 in 2014 and 60 in 2004. The dwindling number of Manhattan’s gas stations, however, is expected to continue, particularly in Lower Manhattan, where real estate prices experience more aggressive growth. To reflect this condition, we assume all gas stations sited in Lower Manhattan, three in total, will go into foreclosure over the course of the project horizon. Gas stations expected to experience foreclosure are itemized as (a) located at 300 Lafayette St., New York, NY 10012; (b) located at 63 8th Ave., New York, NY 10014; and (c) located on FDR Drive & E 23Rd St., New York, NY 10010. The specifications of Manhattan’s gas stations (locations, SR, and brand) and terminals/refineries supplying stations are presented in Appendix A. Tables in Appendix A are adopted from Beheshtian et al. (2017b).

(4) The ratio of the number of en route internal combustion engine vehicles (ICEVs) to the total number of vehicles (ICEVs and electric and fuel-cell vehicles) is expected to drop over time. NYC also expects aggressive projections for market share penetration of electric and emission-free vehicles: a 100-fold increase from the 2014 sales rate, by year 2025 (Electric Vehicle Advisory Committee Report: 2016 Recommendations). Accordingly, we assume the market share of ICEVs drops in time.

(5) Consumer anxiety in times of disaster elevates fuel demand, resulting in hazard-triggered psychological stress (Redlener and Reilly, 2012). Therefore, we assume the ratio of (a) the fuel demand by ICEVs in time of extremes to (b) the demand in fuel during the state of business-as-usual, varies under different scenarios. Assumed numbers are adopted from Beheshtian et al. (2018).

The predefined uncertainties are reported in Table 1.

Table 1 Pre-defined possibilities under alternative disruptive events.

Possibility 2020s 2050s 2080s

100-year flood 500-year flood 100-year flood 500-year flood 100-year flood 500-year flood Descriptor (occurrence likelihood)

Terminals SR 1 0.6 (0.5) 0.8 (0.5) 0.5 (0.5) 0.8 (0.5) 0.4 (0.5) 0.7 (0.5) 2 1 (0.5) 1 (0.5) 0.8 (0.5) 1 (0.5) 0.7 (0.5) 0.9 (0.5)

Gas stations SR 1 0.8 (0.5) 1 (1) 0.7 (0.5) 1 (1) 0.6 (0.5) 1 (1) 2 1 (0.5) – 0.85 (0.5) – 0.8 (0.5) –

Foreclosed gas station (s) 1 a (0.1) a (0.1) a (0.05) a (0.05) a (0.02) a (0.02) 2 b (0.1) b (0.1) b (0.05) b (0.05) b (0.02) b (0.02) 3 c (0.1) c (0.1) c (0.05) c (0.05) c (0.02) c (0.02) 4 a, b (0.1) a, b (0.2) a, b (0.15) a, b (0.15) a, b (0.1) a, b (0.1) 5 a, c (0.2) a, c (0.2) a, c (0.15) a, c (0.15) a, c (0.1) a, c (0.1) 6 b, c (0.2) b, c (0.2) b, c (0.15) b, c (0.15) b, c (0.1) b, c (0.1) 7 a, b, c (0.2) a, b, c (0.1) a, b, c (0.4) a, b, c (0.4) a, b, c (0.64) a, b, c (0.64)

ICEV market share (%) 1 0.9 (0.5) 0.9 (0.5) 0.8 (0.5) 0.8 (0.5) 0.7 (0.5) 0.7 (0.5) 2 0.85 (0.5) 0.85 (0.5) 0.6 (0.5) 0.6 (0.5) 0.5 (0.5) 0.5 (0.5)

Fuel demand 1 1.3 (0.4) 1.5 (0.4) 1.3 (0.4) 1.5 (0.4) 1.3 (0.4) 1.5 (0.4) 2 1.5 (0.6) 1.75 (0.6) 1.5 (0.6) 1.75 (0.6) 1.5 (0.6) 1.75 (0.6)

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Under each of the 100-year flooding events, five lines of uncertainties with 2–7 possibilities for each are defined. This leads to 112 (i.e. 2 possibilities for terminal SR × 2 possibilities for gas station SR × 7 possibilities for foreclosed stations × 2 possibilities for ICEV market share × 2 possibilities for fuel demand = 112) possible sub-scenarios. Every sub-scenario represents an individual occasion associated with a unique set of deterministic input parameters and a likelihood of occurrence. The number of sub-scenarios under 500-year flooding events is 56, half of the sub-scenarios defined for 100-year events, since, there is only one possibility for gas station SR.

The sub-scenarios clustered under a given scenario share the same transportation network topology. Fig. 3 (left) represents the case study area with the transportation network (including 1268 bidirectional arcs and 348 nodes) super-imposed, and the MFSC (including all gas stations located in Manhattan, 29 in total (Table A1), in addition to 18 supplying terminals/refineries (Table A2), all located out of the borough). This figure (right) also shows how a given area is possibly vulnerable against the 100- and 500-year flooding events expected in the 2020s, 2050s, and 2080s.

As discussed in Section 3.2, the vulnerable elements of infrastructure are subject to resilience-enhancing investments. We re- classify resilience-enhancing strategies into four categories: (a) reclusive, which means placing the infrastructure’s critical elements at a standoff-distance by relocating them from vulnerable areas (Sheffi, 2005); (b) absorptive, which means enhancing the ability of infrastructure elements to better withstand the exogenous and extreme shocks; (c) adaptive, which means facilitating the ability of an infrastructure to cope with the aftermath of an extreme event in a shorter time and cost-effectively; and (d) restorative, which means escalating the infrastructure’s overall capacity to recovery from extremes promptly.

Table 2 summarizes the RESs recommended by planning drafts, initiatives, and technical reports which have addressed NYC’s flood-resilient infrastructures, specifically post super-storm Sandy. The strategies listed in Table 2 are categorized in the four cate- gories of reclusive-, absorptive-, adaptive-, and restorative-enhancing, addressing the resilience of critical elements in both infra- structures (transportation and MFSC), including fueling stations, terminals and refineries, reservoir tanks, and transportation arcs.

Each of the strategies is tailored to a given scenario or a range of scenarios. There are, for example, six sub-strategies available to enhance the absorptive-capacity of the elements and three restorative-enhancing sub-strategies assisting the MFSC to recover. There is a corresponding dollar value associated with the cost to deploy each and every sub-strategy, reported in Table 2.

Table 3 presents the notations employed in the optimization model, whose formulation follows.

∑ P IVInMin s

s s (15)

∑

∑ = ∀IVIn

P Y R

P Y s

( )

( ) s

n n s

n s

n s

n n s

n s

(16)

The objective function to be minimized, expression (15), shows the probability-weighted sum of the system-wide inoperability

Fig. 3. (left): case study area including the transportation network and the MFSC; (right): projected 100- and 500-year floodplain maps for 2020s (top), 2050s (middle), and 2080s (bottom), for a sample area.

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across all scenarios/years, and constraint (16) defines the inoperability metric, IVIns, as probability-weighted regrets across the β quantile sub-scenarios.

̂= − ∀R V V s n,ns n s

n s (17)

Table 2 Resilience-enhancing strategies.

Strategy Strategy’s target element

Strategy descriptor Strategy’s target extreme scenario(s)

Cost in $ (×1000)

Reclusive (κ) Node Mobile fuel station/tanker (Initiative 9, NYC Special Initiative for Rebuilding and Resiliency, aka SIRR 2013)

All scenarios 80a

Absorptive (ζ) Gas station Protective roofing and reinforced canopies (Local Laws of the City of New York, 2013), elevated/sealed electric devices (FEMA’s Hurricane Sandy Recovery Advisory report on Restoring Mechanical, Electrical, and Plumbing Systems In Non-Substantially Damaged Residential Buildings, 2013), drainage improvements (the NY Rising Community Reconstruction program, aka NYRCR 2014), deployable floodwalls (Tsvetanov and Shah, 2013) and sandbags (Koch, 2010), water pumps (recommendation 44b: protect subgrade fuel pumps from flooding, FEMA’s Mitigation Assessment Team report, 2013), and shatter-resistant operable windows and frames (Design Guide for Improving Critical Facility Safety from Flooding and High Winds, FEMA’s Risk Management Series, 2007).

2020s 100-y flood 17*,b

2020s 500-y flood 25*

2050s 100-y flood 25*

2050s 500-y flood 40*

2080s 100-y flood 40*

2080s 500-y flood 60*

Terminal Hardening exposed shorelines with armor stone (SIRR, 2013) and constructing levee (New York City Economic Development Corporation, aka NYCEDC 2014), rip-rap and floodwalls (Project EC14-005 2014, Flood Mitigation Engineering Resource Center). Backflow-prevention (Aerts et al., 2013) and sealed devices (Fuel Reports 2013, Association for Convenience and Fuel Retailing), drainage improvements, and elevated critical buildings (New York State Petroleum Terminal Resiliency Assessment).

2020s 100-y flood 12,000**,c

2020s 500-y flood 15,000**

2050s 100-y flood 15,000**

2050s 500-y flood 20,000**

2080s 100-y flood 20,000**

2080s 500-y flood 25,000**

Arc Flood-proofed bridges and tunnels (North Atlantic Coast Comprehensive Study, aka NACC 2015), raised street stocking level (Economics and Statistics Administration, ESA 2013), and tree pruning (National Association of Convenience Stores, aka NACS 2013).

2020s 100-y flood 400***,d

2020s 500-y flood 750***

2050s 100-y flood 750***

2050s 500-y flood 1000***

2080s 100-y flood 1000***

2080s 500-y flood 1200***

Adaptive (δ) Gas station Backup generator and switching key (Gas Station Back-Up Power Program, NYSERDA PON 2758).

All scenarios 23*,e

Terminal Standby emergency generators (SIRR, 2013), backup facilities, and on-site reservoir tanks (NYS Governor Announcement, 2014).

All scenarios 8000**

Restorative (ρ) Node 0.5x106 gallon reservoir tank (New York State Energy Emergency Plan, 2016)

All scenarios 5000f

1x106 gallon reservoir tank All scenarios 7500 3x106 gallon reservoir tank All scenarios 10,000

* Per gas station. ** Per terminal or refinery unit. *** Per mile of single-lane, two-way road. a Estimated cost is adopted from Commercial Truck Trader. b Estimated costs are adopted from the Improvement Center, available at http://www.improvementcenter.com/roofing/3-roofing-materials-that-

can-survive-high-winds.html. c The costs of building and maintaining ‘defense’ against rising sea levels depend on several factors including type (i.e. levee, seawall, rip-rap,

etc.) and height of the structures. To estimate the construction cost, we adopted numbers from several reports/studies: (i) Greater New Orleans Hurricane and Storm Damage Risk Reduction System Facts and Figures (2012), (ii) California Climate Change Center report on defense cost, funded by the Lawrence Berkeley National Laboratory, the University of California, the National Center for Atmospheric Research and the California Air Resources Board (Herberger et al., 2009), (iii) Planning the 'Ike Dike' Defense (Casselman, 2009), (iv) ‘Cost of Defending’ in Mid-Atlantic me- tropolitan coastal areas (Koch, 2010), and (v) CEIWR-HEC 2014, Key Flood Risk Management Terms, US Army Corps of Engineers, Institute of Water Resources. Planning-level cost information for a coastal levee designed for a 1 percent chance (100-year) event is reported by USACE (2013f)

d Estimated costs are adopted from (i) Flood Impact Assessment Report Arapaho and Roosevelt National Forests and Pawnee National Grassland October (2013), (ii) Flood Rapid Assessment Model (F-RAM), developed by the California Department of Water Resources (DWR), (iii) Strategies for Flood Risk Reduction for Vulnerable Coastal Populations around Delaware Bay 2014, developed by New Jersey Governor’s Office of Recovery and Rebuilding and New Jersey Department of Environmental Protection, (iv) The Economic Cost of Sea Level Rise to Three Chesapeake Bay Communities, 2003, developed by Maryland Department of Natural Resources, Climate Change in the United States: Benefits of Global Action 2015, developed by EPA, (v) Climate, Sea Level Rise and Planning for the Future 2017, and vi) Emergency Tree Risk Management in NYC report (2012).

e NYS Governor Announcement (2013), available at https://www.governor.ny.gov/news/governor-cuomo-announces-more-250-downstate-gas- stations-installing-back-power-capacity-prepare

f Fuel NY Initiative, Governor's Office of Storm Recovery (GOSR 2013).

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Table 3 Nomenclature.

Sets

{y} = {2020s, 2050s, 2080s} Year {s} = {1,2,3,…, S}; S = 6 Scenario {n} = {1, 2, …, N}; N = 504 Sub-scenario {r} = {κ, ζ, δ, ρ } Resilience-enhancing strategy {ter} = {1, 2, …, TER}; TER = 18 Supply node (terminal/refinery) {t} = {1, 2, …, T}; T = 348 Transshipment node (tt for alias) {t’} = {1, 2, …, T'}; T’=348 Transshipment node on auxiliary network* (i, j, and tt' for alias) {d} = {1,2, …, D}; D = 29 Demand node (gas station) {e} = {TER, D, T, (T,TT), (TER,T),

(T,D)} Network element

Parameters

Ps Arrival likelihood (occurrence chance) of scenario s Pn

s Arrival likelihood (occurrence chance) of sub-scenario n

̂Vn s The maximum performance system can possibly pose during sub-scenario n, scenario s

αs Constant α, pre-assigned for scenario s βs Constant β, pre-assigned for scenario s IPRr e

s , Binary input parameter: 0 if strategy r is designed/needed to enhance the resilience of element e during scenario s, 1

otherwise IPRt Binary input parameter: 1 if location t is a candidate location to host a reservoir; 0 otherwise IPOe

s n, Binary input parameter: 1 if element e during scenario s and sub-scenario n is operable, 0 otherwise

IPMt s n, Binary input parameter: whether node t is a candidate to host a fuel tanker

Wr e ss t ,

, Capacity of the auxiliary network’s arc connecting super-supply ss to node t and related to RES r

Wr e t tt , , Capacity of the auxiliary network’s arc connecting two nodes and related to RES r

We s n, Maximum capacity of element e during sub-scenario n, scenario s. The element’s flow capacity is a function of the

element’s coordinate and whether it sits in any of the flood-prone areas assigned to sub-scenario n, scenario s M Scalar larger than 1 FVt j

tt t , , Fuel capacity (in gallon) of the reservoir type j to be sited in node t

Wt t sd, Flow capacity of the auxiliary network’s arc connecting node t to supper-demand sd

Wt t tt, Flow capacity of the auxiliary network’s arc connecting two nodes

Cr e i j , , Cost of the RES r invested in element e

mr Marginal interest rate CRt j Unit price of the reservoir tank type j CTk Unit price of the fuel tanker Tk cap Capacity of the fuel tanker (in gallon)

Variables

IVIns The binary index of the system-wide inoperability during scenario s Rn

s System-wide regret corresponding to sub-scenario n, scenario s

Vn s The system performance during sub-scenario n, scenario s, recommended by solution of the model

Xt d s n , , Fuel delivered in gas station d from node t, during sub-scenario n, scenario s

VMt s n, Fuel delivered in node t via mobile tankers and during sub-scenario n, scenario s

Zn s Binary decision variable whether sub-scenario n in scenario s is clustered in α quantile

Yn s Binary decision variable whether sub-scenario n in scenario s is clustered in β quantile

φs The maximum inoperability of any α quantile sub-scenarios of scenario s IVe

s n, Binary decision variable whether element e is fully robust during scenario s, sub-scenario n

IVFr e s , Binary decision variable whether RES r on element e is taken place on right time/level to protect the element during

scenario s

IVNr e i j , , Binary decision variable whether investment on RES r is taken place on element e

IVOpe s n, Binary decision variable whether element e is resilient during scenario s, sub-scenario n

Yr e ss t ,

, Binary investment decision (flow on auxiliary arc) corresponding to element e and RES r

Yr e t sd , , Binary investment decision (flow on auxiliary arc) corresponding to element e and RES r

Yr e t tt , , Binary investment decision (flow on auxiliary arc) corresponding to element e and RES r

Yr e i j , , Binary investment decision (flow on auxiliary arc) corresponding to element e and RES r

Xt tt s n , , Physical flow (in gallon) between transshipment nodes, during scenario s, sub-scenario n

Xt d s n , , Physical flow (in gallon) delivered in gas station d from node t, during scenario s, sub-scenario n

Xter t s n

, , Physical flow (in gallon) supplied by terminal ter and shipped to node t, during scenario s, sub-scenario n

VRTt s Fuel (in gallon) supplied by reservoir at node t, during scenario s

XRtt i j, Binary decision variable on auxiliary network indicating whether reservoir is funded/sited on node t

XRtt t sd, Binary decision variable on auxiliary network corresponding to node t and super-demand sd

(continued on next page)

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∑ ∑= + ∀V X VM s n,ns t d

t d s n

t t s n

, , , ,

(18)

Constraint (17) reflects the regrets corresponding to each of the sub-scenarios. This equals the difference between the maximum operability level the infrastructure may possess during sub-scenario n, given no cap on the funds available for infrastructure in- vestment ( ̂V )n

s and the system’s performance to be optimized by the model (Vn

s). Constraint (18) defines the operability index, V ,n s the

summation of (1) total fuel supplied by terminals and distributed across gas stations and (2) physical flow supplied by mobile tankers.

∑ ⩾ ∀P Z s( ) α n

n s

n s s

(19)

∑ ⩾ ∀P Y β s( ) n

n s

n s s

(20)

≤ ∀R Z φ s n,n s

n s s (21)

⩾ ∀R Y φ s n,n s

n s s (22)

Constraints (19)–(22) reflect the reliability aspect of the model as discussed in Section 3.4. Constraints (19), (20) assure the cumulative occurrence chances of the sub-scenarios clustered under each quantile are larger-equal the pre-assigned values of α and β. Constraints (21), (22) eliminate the regret corresponds to each and every sub-scenario clustered under α quantile to exceed the regret corresponding to any of the β quantile sub-scenarios.

≤ ∀X W IVOp e s n, ,e s n

e s n

e s n, , ,

(23)

= + − ∀IVOp IPO IV IPO e s n(1 ) , ,e s n

e s n

e s n

e s n, , , ,

(24)

∏= + − ∀IV IPR IVF IPR e s n( (1 ) ) , ,es n R

r e s

r e s

r e s,

, , , (25)

Constraints (23)–(25) condition the operability of the vulnerable elements on investment in corresponding RESs. Constraint (23) governs the operability of the element e and restricts its functionality during sub-scenario n (whether Xe

s n, can be larger than zero) to the element’s maximum-nominal capacity denoted by We

s n, . While We s n, is an input parameter, binary variable IVOpe

s n, is designed to relax the element’s capacity if it is inoperable during sub-scenario n. Constraint (24) conditions the operability of element e during sub-scenario n on either the inherent robustness of the element against sub-scenario n’s corresponding aftermath (when binary input parameter IPOe

s n, holds value 1) or implication of the RESs that can handle the corresponding shock (requiring the binary decision variable IVe

s n, to gain value 1, if the model decides to enhance the resilience of the element e in the face of sub-scenario n). Constraint (25) conditions the enhanced-resilience of element e during sub-scenario n on investment in those resilience-enhancing

strategies (i.e. κ, ζ, δ, and ρ) which are designed for this element. Within this constraint, the term ∏ +IPR IVF(R r e s

r e s

, , (1-IPRr e s , )) is a

dummy variable assuring all RESs are considered. To hold value 1, this binary variable conditions element e’s resilience-enhancement on (1) availability of the RES r to enhance element e’s resilience during scenario s (see Table 2) and (2) whether investment in RES r on element e has taken place on the right time/level required to protect the element during scenario s.

∑ ⎜ ⎟= ⎛ ⎝

− + − ′ ⎞ ⎠

∀IVF IVN j s ε s j

e s r 6 6

) , ,r e s

i j r e i j

, ,

, ,

(26)

∑ = ∀Y e r1 , t

r e ss t ,

,

(27)

∑ = ∀Y e r1 , t

r e t sd , ,

(28)

Table 3 (continued)

Sets

XRtt ss t, Binary decision variable on auxiliary network corresponding to node t and super-supply ss

CT Project’s total cost across all scenarios and all years, normalized on present dollar value (2017) C y Project’s total cost in year y CRT y Cost spent on reservoir(s) in year y CRT j Cost of reservoir type j CTky Cost spent on tanker(s) in year y Ce

y Cost spent on RESs of all the elements in year y

NTkY Number of tankers deployed in year y NTk s Number of tankers deployed during scenario s NTk s n, Number of tankers deployed during sub-scenario n, scenario s

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∑ ∑ ∑+ = + ∀′Y Y Y Y e r, t

r e ss t

tt r e t tt

t r e tt t

r e t sd

, ,

, ,

, ,

, ,

(29)

⩾ ∀IVN Y e r i j, , ,r e i j

r e i j

, ,

, ,

(30)

≤ ∀IVN MY e r i j, , ,r e i j

r e i j

, ,

, ,

(31)

Constraints (26)–(31) relate to decision variables reflecting the investment of RESs in vulnerable elements. Constraint (23) assures the RES type r has taken place at the right time/level. Constraints (27)–(31) correspond to an auxiliary network which is designed to reflect the finance piece of the model, as further elaborated in Appendix B.

∑ ∑ ∑ ∑ ∑+ − − ≤ + ∀ ′ ∈ ′ ≤ ′

X X X X IPR VRT IPM VM n s s S s s( ) , ( , ) tt

t tt s n

d t d s n

tt tt t s n

ter ter t s n

s t t

s t s n

t s n

, ,

, ,

, ,

, , , ,

(32)

∑ ⎜ ⎟⎜ ⎟= ⎛ ⎝

⎛ ⎝

− ″ − −

⎞ ⎠

+ ⎞ ⎠

∀VRT XRt FV s j

ε s t | | 0.5

10 ,t

s

i j t i j

t j tt t

,

, ,

,

(33)

∑ ∑+ = + ∀′ ′ ′ ′ ′XRt XRt XRt XRt e r,tss t tt

t t

t t tt t

t t sd, ,tt , ,

(34)

∑ = ′

′XRt 1 t

t ss t,

(35)

∑ = ′

′XRt 1 t

t t sd,

(36)

Constraints (32)–(36) locate reservoir tanks and mobile fuel tankers in transshipment nodes. Constraint (32) is a flow con- servation constraint at node t matching inflow, outflow, and potential supply (i.e. fuel available either through reservoir tanks, denoted by ∑ ′ IPR VRT( )s t t

s , or mobile tankers, denoted by IPM VMt s n

t s n, , ). Expression ∑ ′ IPR VRT( )s t t

s , a dummy variable, represents reservoir tank capacity in transshipment node t suggested by the model for any scenario ′s no larger than s. The number of tanks in one location is limited to one.

Constraint (33) represents available fuel (in gallons) at node t to be supplied by reservoir tank if the tank is selected by the model to be sited in this node (i.e. if binary variable XRtt

i j, = 1). Constraint (34) corresponds to flow conservation for transshipment nodes on an auxiliary network, and constraints (35), (36) show supply and demand on super-supply (SS) and super-demand (SD) nodes in an auxiliary network, respectively. Auxiliary network and input parameters related to constraints (33)–(36) are elaborated in the Appendix B.

∑ ⎜ ⎟= ⎛ ⎝

+ + +

⎞ ⎠

∀ −

CT C CRT CTk

mr y

(1 )y

y y y

y y0 (37)

≤CT BC (38)

∑= ∀C C yy e

e y

(39)

∑ ⎜ ⎟= ⎛ ⎝

⎡ ⎣

− − ′ + ⎤

⎦ ⎞ ⎠

∀C IVN C y s

y e | | 65

1 ,e y

i j r r e i j

r e i j

, , , ,

, ,

(40)

Constraints (37)–(40) reflect the project’s expected cost across the years, and show how the time-distributed investment cost converts to present dollar value. Applying the mr factor, constraint (37) converts the total investment value for different network elements and over the years to the present dollar value. Constraint (38) assures the investment value does not exceed the project’s budget cap. Constraint (39) represents the investment values for all elements in time y, and constraint (40) adds up the dollar values of all RESs invested in element e, time y. In this constraint, expression +− − 1y s| |

65 , a binary variable, matches scenario s with the time

it is expected to occur in year y. Given the input parameters listed in Appendix B, for instance, if Ce y is the cost to be invested in

element e in the 2020s, the binary variable to hold value 1 should have l s( ) = 20 meaning only scenario 1 and 2 could be considered for Ce

2020.

∑ ⎜ ⎟= ⎛ ⎝

⎡ ⎣

− − ′ + ⎤

⎦ ⎞ ⎠

∀RT y s

XRt CRT yC | | 65

1y t i j

t i j j

, ,

,

(41)

Similar to constraint (40), constrain (41) reflects the cost related to reservoir tank(s) to be invested in time y.

= ∀CTk NTk CTk yy y (42)

⩾ − − ′

+ ∀NTk NTk y s

ε s y | | 65

,y s (43)

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⩾ ∀NTk NTk s n,s s n, (44)

∑= ∀NTk VM Tk

s n,s n t

t s n

cap ,

,

(45)

Constraint (42) reflects the investment in year y required for obtaining an NTky number of tankers. Constraint (43) links the number of tankers required in year y to the number of tankers selected by the model for scenario s. Constraint (44) assures the number of tankers assigned to scenario s is adequate during all scenario s’s sub-scenarios. Constraint (45) assures the number of tankers selected for sub-scenario n, NTk s n, , is enough to distribute physical flow to be distributed by mobile tankers.

5. Numerical analysis and implications

The model is framed within the mixed integer quadratically constrained program (MIQCP), and coded in the general algebraic modeling system (GAMS). To run the code, we utilized a Lenovo ThinkStation P910 equipped with a dual-processor, dodeca-core, and 128 GB DDR4 SDRAM. The Convex Over and Under ENvelopes for Nonlinear Estimation (COUENNE) also used as a solver.

Ten numerical experiments have been developed to evaluate the modeling results under different input parameters and through different objective functions. These experiments are listed in Table 4. The first five experiments explore the proposed model through different input values assigned to α, β, and mr. Experiments 6–8 represent α-reliable and its extensions: α-reliable worst-case and α- reliable mean-excess. Experiment 9 represents the generic (i.e. weighted-average) objective function, and experiment 10 simulates minimax objective function. For the sake of brevity, we graph the result of the first experiment in detail (Fig. 4) and later will draw a comparison between the outputs of all experiments.

Under no investment plan (i.e. when the MFSC faces climatic extremes while deprived of resilience-enhancing strategies), the infrastructure experiences considerable loss on system-wide operability. The sub-scenario-specific levels of operability are re- presented in Fig. 4, red line. The infrastructure is subject to 95.1% of expected inoperability averaged over all sub-scenarios and across the project’s time horizon (i.e. 4.9% operability), if no resilience-enhancing investment financed (Fig. 4, lower dashed-line). In the aftermath of the extremes assumed during 500-year scenarios, however, the average drop in system operability is 57.2% more

Table 4 Experiments descriptor.

Experiment Objective function Input parameters Risk to be tolerated mr (%) Running time (sec)

α β

1 Proposed model 0.8 0.05 0.15 6 79 2 Proposed model 0.8 0.05 0.15 4 79 3 Proposed model 0.8 0.05 0.15 2 80 4 Proposed model 0.8 0.1 0.1 6 76 5 Proposed model 0.8 0.15 0.05 6 77 6 α-reliable 0.8 – 0.2 6 65 7 α-reliable, worst-case 0.8 – 0.2 6 62 8 α-reliable, mean-excess 0.8 – 0 6 72 9 Weighted-average – – 0 6 83 10 Minimax – – 0 6 61

Fig. 4. Enhanced-operability under experiment 1.

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than the average drop under 100-year scenarios. The most sever drops in system operability are observed in those sub-scenarios representing minimum supply (due to either terminals dropped SR or foreclosed gas stations) and maximum demand (due to e.g. consumer anxiety).

On the other hand, if resources are fully allocated on the sub-scenarios, individually (i.e. running the model for the maximum resilience the MFSC may possibly experience through each of the sub-scenarios), the infrastructure’s operability reaches up to an average of 21.62% (Fig. 4, upper dashed-line). This is analogous to the averaged values of ̂Vn

s across all sub-scenarios (Fig. 4, black

solid line). Full allocation of resources, however, induces the largest resilience-enhancement on those sub-scenarios which experi- enced most the severe drop on operability.

Also, the average marginal difference between maximum ( ̂Vn s ) and minimum operability for the 2020s sub-scenarios is 26.7% and

39.1% larger than the same margin for the 2050s/2080s sub-scenarios. This is due to the fact that the vulnerability of the 2050s/ 2080s sub-scenarios are multi-sources, hence the model-given capped budget has a deficient effect on restoring system resilience during more severe events. For example, under a number of sub-scenarios assigned to the 2080s’ 500-year flooding (i.e. the most severe extremes simulated in this study), the MFSC experiences full shutdown (i.e. operability = 0) caused by inoperability of 83% of terminals/refineries, closure of the entire points of entry (12 bridges and tunnels, in total) connecting Manhattan to the rest of the case study areas, flooded service stations, etc. Therefore, the project’s restricted budget has either limited or no influence on en- hancing the infrastructure’s operability.

The area between the red line (system’s inherent operability in time of disaster) and the black line (highest operability the system may possibly possess following investment) represents the range of operability-enhancement the MFSC may experience during each sub-scenario. Solving for minimum weighted-average regret, the optimum allocating of resources improves the system’s average operability by 7.19%. This means a set of optimum investment decisions provides the system with 12.09% operability averaged over all sub-scenarios (middle dashed-line). The area colored in light gray is the system-wide operability improved by the implementation of those investment decisions suggested by the model and the area colored in dark gray represents the regret remaining after the investment. The weighted average of the dark gray area over the sub-scenarios’ likelihood denotes the least possible regret.

We further discuss the model outputs and analyze (1) time-distribution of investments, (2) model response to the severity of extreme events, (3) the impact of sub-scenarios on investment decisions, (4) the pattern of resource allocation on different resilience- enhancing strategies, and (5) the difference across the experiments.

5.1. Time-distribution of investments

The proposed model is designed to optimally allocate assets/resources across the modeling time horizon. This means the model addresses the two main questions of when to invest and in which strategy to invest. Accordingly, the proposed model performs an intertemporal choice, a tradeoff in three principle areas between (a) early investment: taking the risk of vulnerable infrastructure in the face of intensive events in the mid- and long-term, and moderate events in the long-term, (b) mid-term investment: staying vulnerable to short-term hazards and intensive long-term events, and (c) late investment: taking the risk of low-resilient infra- structure in the face of climatic extremes happening in short- and mid-term.

A large share of resources is invested in the 2020s (i.e. early investment of 69.3% of the available budget), yet only 20.8% and 9.9% of the available budget are assigned for 2050s and 2080s investments, respectively. The reasons are twofold: (1) postponing investment to the 2050s/2080s will provide the system with small marginal resilience compare to what the system may experience due to early investment, (2) early investments are not specifically designed for long-term resilience, however, may partially enhance system resilience in the face of 2050s and 2080s extremes. Resulting from such investments, the average enhanced-resilience during the 2020s’ sub-scenarios is 26.6% and 50.3% larger than the same metrics corresponding to the 2050s and 2080s sub-scenarios, respectively.

The time-distribution of investments, however, varies under experiments 2 and 3 where the marginal rates of interest are lower. Through the second experiment (mr = 4), the model allocates 59.4%, 26.5%, and 14.1% of the available budget in the 2020s, 2050s, and 2080s strategies, respectively. Under experiment 3 (mr = 2), these values are 58.8%, 26.0%, and 15.2%. While the interest rates for 2020s-2080s are significantly different, the model doesn’t suggest an aggressive investment strategy for the 2050s and 2080s. The reason lies in the limited role of investment (at-least under the assumed budget cap) in infrastructure resilience-enhancement when extremes are as severe as the ones expected for the 2050s and 2080s.

5.2. Model response to the severity of extreme events

While the likelihood of the 500-year extremes is 20% of the 100-year extremes (i.e. they have less weight in the objective function), the average resilience-enhancement of the sub-scenarios under 500-year events is higher than the same index for the 100- year sub-scenarios. The investments in RESs are exclusively assigned to a series of (sub)-scenarios clustered under β quantile, however, a given strategy may help the MFSC to better absorb, adapt to, or cope with a range of sub-scenarios not included in β quantile.

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The correlation between the level of resilience-enhancement against and the severity of the extreme events follows different patterns across the time horizon. The averaged resilience improvements assigned to the 500-year’s sub-scenarios of each time- window drops in time. This is due to the high severity and the sub-scenarios’ low ̂Vn

s corresponding to the 2050s and 2080s extremes.

Through the second and third experiments (shown in Fig. 5), where lower mrs are supposed to make the late investments more reasonable, the drops in the marginal resilience-enhancements across the time-frames are less considerable. Enhanced-operability through the first three experiments are compared in Fig. 5.

5.3. Impact of sub-scenarios on investment decisions

During a given sub-scenario and due to a set of investments, the MFSC experiences some degree of enhanced-resilience. The enhanced-resilience, as well, depends in part on the infrastructure’s attributes pre-defined earlier for the sub-scenario. The resilience- enhancement on those sub-scenarios corresponding to significant vulnerability (Fig. 4, red line) is larger than the resilience improved on sub-scenarios experiencing less disruption. Through the first experiment, over 60% of the weighted-average, enhanced operability relates to those sub-scenarios with the lowest inherent resilience (33% of the sub-scenarios), despite the fact that a considerable portion of these sub-scenarios (i.e. 33–15 = 18%) are not the target of the model (i.e. those sub-scenarios do not fall in β quantile).

The reason lies in modeling properties that sub-scenarios share with each other. For example, the modeling suggestion to invest in a reservoir tank increases the infrastructure well-being under all sub-scenarios, despite the fact that a given strategy is deployed to only minimize the regrets associated with the β quantile sub-scenarios. Through the second and third experiments, the model results are close to the results for the first experiments, however, the investment decisions are slightly postponed.

5.4. Pattern of resource allocation on different resilience-enhancing strategies

Comparing the results of the experiments listed in Table 4, Fig. 6 represents how the model distributes resources across different strategies (i.e. κ, ζ, δ, and ρ) and various elements of the infrastructure (i.e. tanker, reservoir, gas station, road, and terminal).

Experiments 1–5 simulate risk-tolerating conditions, while moderate risk-seeking and risk-aversion scenarios are modeled through experiments 6 and 7, respectively. By increasing the risk-aversion state, the model covers those sub-scenarios in which the

Fig. 5. Enhanced-operability through experiments 1–3.

Fig. 6. Asset allocation across the MFSC’s elements (left) and across different types of strategy (right).

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regrets are more significant. To this end, the model shifts resources from restorative- to absorptive-enhancing strategies. In fact, in risk-averse state, more resources are shifted towards absorptive-enhancing investments which are mostly on transportation arcs (i.e. strategies such as flood-proofed bridges and tunnels, and tree pruning). Whereas, within the risk-tolerance state, resources are mostly invested on reservoir tanks through the restorative-enhancing strategies.

Under the late response investment options, as modeled in experiments 2–3, the model allocates resources on network elements and strategy types differently. When mr drops, the late investments are shifted to absorptive-enhancing strategies on transportation arcs. This means the optimum investment decisions support reservoir tanks through the early/2020s investment, since this type of strategy is not assumed to be sub-scenario-specific and operational during all extremes. Nevertheless, through experiments 2–3, late- investment is mainly concentrated in the resilience-enhancement of bridges/tunnels. Another interesting result relates to the com- parison of experiments 7 and 10, where two minimax regret models are analyzed. In both experiments, the model focuses on sub- scenarios with the lowest regret values within the quantiles of interest (i.e. 0.8 quantile in experiment 7 and 1 quantile in experiment 10). While experiment 7 possesses a risk-tolerant state (compared with experiment 10), where more resources are invested on restorative-enhancing strategies (the same trend observed across the first three experiments) and fuel being distributed, mainly, by mobile gas stations (tankers).

5.5. Experiments comparison

We summarize and compare some features of the experiments in Table 5. Through the first three experiments, by dropping the mrs, the averaged enhanced-resilience for sub-scenarios under all quantiles slightly increases. This is due to the larger present dollar value through experiments 2–3, meaning the model has a larger amount of funds to invest in strategies. Also, while the model is assigned to exclusively minimize the regrets corresponding to β quantile sub-scenarios, the investment impacts the α quantile sub- scenarios more significantly, whereas, the 1-α-β quantile sub-scenarios experience resilience enhancement to a lower degree. The reason for this relates to the complexity in resilience enhancement of the β quantile sub-scenarios, which demand stronger strategies compared to those strategies required for the α quantile sub-scenarios. The averaged resilience-enhancement over all sub-scenarios also improves when mrs drop.

Through experiments 6–8 which represent different extensions to the α-reliable objective function, the MFSC experiences higher resilience. The average resilience through experiment 7, α-reliable worst-case, is lower than the resilience driven through other extensions since the model concentrates on a limited number of sub-scenarios. In theory, the model minimizes the regret corre- sponding to a single sub-scenario. To do so, nevertheless, the model allocates resources on at least one sub-scenario and at most on 100α% of the sub-scenarios. Therefore, the averaged resilience-enhancement on the system is beyond the impact of one resilient sub- scenario. Not surprisingly, the maximum averaged resilience is driven through experiment 9 where the objective function reflects the weighted average over all sub-scenarios.

6. Conclusion

The complexity inherent in infrastructure of transportation energy within the context of a growing frequency and intensity of climatic hazards requires improvements in decision-making capacities and long-term planning response solutions. The existing ap- proaches toward resilient fueling infrastructure, however, come with two major shortcomings: the lack of system thinking to enhance the resilience of infrastructure and the absence of intertemporal decision-making mechanisms. In this research, we proposed a modeling framework to study the long-term resilience of Manhattan’s motor fuel supply chain in the face of (1) gradual, permanent flooding imposed by sea-level-rise and (2) extreme-sudden flooding events.

Table 5 Experiments results.

Exp. Averaged enhanced operability (%)

α quantile β quantile 1-α-β quantile Entire sub-scenarios

1 10.4 7.39 5.59 9.53 2 10.68 7.42 5.62 9.76 3 10.71 7.41 5.66 9.78 4 10.91 7.16 5.86 10.03 5 11.06 7.01 5.83 10.19 6 18.6 – – 14.1 7 – – – 10.5 8 – – – 12.9 9 – – – 19.7 10 – – – 6.2

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The results from our study indicate that different allocations of assets through a variety of objective functions can lead to sig- nificant differences in post-disaster operational decisions for climate-vulnerable energy systems. An alternative to the risk averse and risk neutral models, the proposed hybrid utility function increases relative risk aversion, yet reduces the absolute risk aversion needed to avoid those sets of predictions for low-payoff treatments. The former property of the proposed hybrid model (i.e. risk aversion) protects the infrastructure against a range of likely, high impact hazards which are ignored by risk-tolerant management strategies. While, the latter property (i.e. risk neutrality) increases the infrastructure operability in time of low/medium size extremes (analogues to risk neutral models which lead to significant improvement in the system’s averaged resilience). This thus helps pol- icymakers to weight in (1) non-structural strategies and (2) those structural, yet non-adaptive strategies (e.g. post-disaster rapid response, long-term reconstruction funds, etc.) to substitute climate-adaptive physical investments, and accordingly, secure financing for non-substitutable physical investments.

Despite ongoing proclamations that municipalities and governments should focus their adaptation fund on supporting the de- ployment of later-stage solutions, the model suggests an intertemporal portfolio to be financed across the project’s time horizon. Even under lower marginal rates of interest, the model suggests investments in early- and late-stage solutions as a complementary ap- proach with significant weight on immediate actions.

The model, under all experiments and objective functions, supports a decentralized supply chain structure. The model neutralized the role of the infrastructure’s key-bottlenecks (i.e. Manhattan’s tunnels/barges) in the system’s post-disaster operability by deploying reservoir tanks within the borough of Manhattan, a solution that has never been considered by city/state authorities. Such invest- ment, an immediate-early stage decision as suggested by the model, should be considered as a no regret, yet non-zero-sum strategy that benefits NYC in all possible future climate circumstances.

There are several recommendations to improve and further investigate the properties of this hybrid model. First, modeling the risk/return plot and developing Pareto-optimal portfolios can improve the finance piece of the model. As well, one may advance the ‘economics of adaptation’ by inclusion of the Stackelberg leadership model to provoke the investment and shift the ‘burden/direct rate’ on private sectors benefiting from the resilience-enhanced infrastructures. Second, as frequently highlighted by institutional NGOs (e.g. the Global Facility for Disaster Reduction and Recovery of the World Bank Group and the United Nations Office for Disaster Risk Reduction), “non-structural” measures matter. Therefore, land-use policies and regulations, and their enforcement, among other non-physical RESs, may reduce or otherwise neutralize the impact of climate-related hazards.

Third, investment in and successful deployment of climate-adaptive strategies require more studies on the multi-jurisdictional nature of the New York metropolitan area which stretch across New York, New Jersey, and Connecticut. For example, many of NYC’s bridges and tunnels cross state lines and/or fall under the control of the Port Authority, a joint venture between the New York and New Jersey. Moreover, infrastructure planning is a transboundary problem set. Effective planning, therefore, must devise a wide range of strategies. A successful framework would incorporate long-term deliberative climate-adaptive approaches and immediate disaster risk management into policies for the planning, retrofitting, and reconstruction of those infrastructures critically vulnerable to climatic hazards.

While the proposed model is exclusively developed to address the long-term resilience of Manhattan’s transportation energy infrastructure, the structure of the model could be applied to other NYC’s supply chains. One possible infrastructure is the NYC’s food supply chain whose climate-adaptability is a new concern raised by the NYC Mayor’s Office of Recovery and Resiliency and NYC Economic Development Corporation. Furthermore, the proposed model could be adopted for climate-vulnerable fueling supply chains in other case study areas such as Houston, TX, and Miami, FL, which recently experienced widespread dysfunctionality in the aftermath of hurricanes Harvey and Irma, respectively.

Acknowledgements

Co-author H. Gao acknowledges partial support by NSF project CMMI-1462289, Natural Science Foundation of China (NSFC) project #71428001, and Lloyd’s Register Foundation project agreement ID: 80034. We thank the reviewers for their comments and suggestions, which helped us improve the paper.

Appendix A

See Tables A1 and A2.

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Appendix B

Here, we further elaborate on details of some of the constraints and show their syntax in GAMS.

B.1. Constraint (26)–(31)

Decisions related to optimum allocation of assets (i.e. in what time and on what strategy/element to investment in) are modeled in a secondary/auxiliary graph (assuming the graph represents the networked infrastructure of MFSC a primary one). As shown in the Fig. B1, we assume a general network G = (V, A) for each strategy r and for every element e. Each node (∈V) serves as either a source

Table A1 Manhattan’s gas stations monthly sale rate.

Gas station Monthly sale (gallon) Address Brand

1 155,000 300 Lafayette St./21 E Houston St, New York, NY 10012 BP 2 130,900 2430 East FDR Dr., New York, NY 10010 BP 3 150,000 51 8th Ave., New York, NY 10014 Mobil 4 124,600 309 11th Ave., New York, NY 10001 Mobil 5 155,000 466 10th Ave. New York, NY 10018 BP 6 125,000 502W 45th St. #32215, New York, NY 10036 Hess Express 7 115,000 639 11th Ave., New York, NY 10036 Sunoco 8 144,500 718 11th Ave., New York, NY 10019 Mobil 9 166,700 1855 1st Ave., New York, NY 10128 Shell 10 158,000 303W 96th St., New York, NY 10025 Mobil 11 164,000 1599 Lexington Ave., New York, NY 10029-6122 Shell 12 114,800 348 E 106th St., New York, NY 10029 Getty 13 135,900 2276 1st Ave., New York, NY 10035 Shell 14 156,520 2326 1st Ave., New York, NY 10035 BP 15 140,770 255 E 125th St. (2442 2nd Ave), New York, NY 10035 BP 16 159,440 1890 Park Ave. New York, NY 10035 BP 17 132,870 117 Morningside Ave. (2 Hancock Pl) New York, NY 10027 Shell 18 90,000 3260 Broadway, New York, NY 10027-7923 Shell 19 105,200 150-54 West 145th St., New York, NY 10039 Mobil 20 149,600 232W 145th St., New York, NY 10039 Shell 21 154,900 800 St Nicholas Ave., New York, NY 10031 BP 22 91,700 3740 Broadway, New York, NY 10032 Mobil 23 98,470 2165 Amsterdam Ave., New York, NY 10032 BP 24 105,700 4275 Broadway, New York, NY 10033 Raamco 25 95,290 4353 Broadway, New York, NY 10033 BP 26 132,800 265 Nagle Ave., New York, NY 10034 BP 27 127,000 242 Dyckman St., New York, NY 10034 BP 28 102,380 3936 10th Ave., New York, NY 10034 BP 29 93,250 5080 Broadway, New York, NY 10034 Sunoco

Table A2 Terminals and refineries supply rate.

Terminal/Refinery Address

BP Production North America 125 Apollo St. Brooklyn, NY 11222 BP Carteret Terminal 760 Roosevelt Ave., Carteret, NJ 07008 BP Marine Americas 350 Coastal St., Port Newark, NJ 07114 Mobil at BP Carteret 760 Roosevelt Ave., Carteret, NJ 07008 Mobil – Global Inwood 464 Doughty Blvd., Inwood NY 11096 Hess Corp. 750 Cliff Rd, Woodbridge, NJ 07095 Hess Corporation Bronx Terminal 1040 E 149th St., Bronx, NY 10455 Hess Corporation Brooklyn Terminal 722 Court St., Brooklyn, NY 11231 Hess Corporation, Bayonne 420 New Hook Rd., Bayonne, NJ 07002 Sunoco Logistics Partners LP 436 Doremus Ave., Newark, NJ 07105 Sunoco Inwood Terminal 70 East Ave., Lawrence, NY 11559 Shell/Motiva Brooklyn Terminal 25 Paidge Ave., Brooklyn, NY 11222 Getty – Global Inwood 464 Doughty Blvd., Inwood NY 11096 Getty – Hess Corp. 750 Cliff Rd., Woodbridge, NJ 07095 Getty – BP Production North America 125 Apollo St. Brooklyn, NY 11222 Getty – BP Carteret Terminal 760 Roosevelt Ave., Carteret, NJ 07008 Getty – Hess Corp. 350 Coastal St., Port Newark, NJ 07114 Raamco 760 Roosevelt Ave., Carteret, NJ 07008

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(super-supply, SS), sink (super-demand, SD), or transshipment node (i and j). SS node has supply equal to one unit and SD has demand equal to one unit. SS and SD nodes are connected through a network of arcs (i, j) linking three sets of transshipment nodes each represents six resilience-enhancing, candidate strategies (each strategy is tailored to response the vulnerability imposed by each of the scenarios 1–6) for investment in the 2020s, 2050s, and 2080s.

Flow of one unit departs SS node, gets distributed across transshipment arc (i, j), and sinks in SD node. Each of the arcs carries out a unit capacity and the cost corresponding the physical investment on RES in a particular level. The flow distribution, therefore, represents the decision of planner regarding whether the model decides to invest in a particular RES r on element e, and if it does, optimum timing to do so.

To each of the transshipment nodes, we assign a series of input parameters (Table B1). As shown in constraints (23)–(25), operability of the element e (if the element is not inherently robust) during scenario s is

Fig. B1. Auxiliary network for RES r, element e. Note: grayed boxes represent those RESs which are capable to enhance the element e’s resilience in time of the year y’s assigned extreme events.

Table B1 Corresponding parameters to transshipment nodes in network G.

Node Scenario Extreme event ord(j) ord(s) ord(s’) ord(j’)

SS – – – – – – Dummy – – – – – – 2020-No Inv. – – – – – – 2050-No Inv. – – – – – – 2080-No Inv. – – – – – – 20-S1 1 100-YF, 2020s 1 1 20 0.5 20-S2 2 500-YF, 2020s 2 2 20 0.5 20-S3 3 100-YF, 2050s 3 3 20 0.5 20-S4 4 500-YF, 2050s 4 4 20 0.5 20-S5 5 100-YF, 2080s 5 5 20 0.5 20-S6 6 500-YF, 2080s 6 6 20 0.5 50-S1 1 100-YF, 2020s 1 1 50 2.5 50-S2 2 500-YF, 2020s 2 2 50 2.5 50-S3 3 100-YF, 2050s 3 3 50 2.5 50-S4 4 500-YF, 2050s 4 4 50 2.5 50-S5 5 100-YF, 2080s 5 5 50 2.5 50-S6 6 500-YF, 2080s 6 6 50 2.5 80-S1 1 100-YF, 2020s 1 1 80 4.5 80-S2 2 500-YF, 2020s 2 2 80 4.5 80-S3 3 100-YF, 2050s 3 3 80 4.5 80-S4 4 500-YF, 2050s 4 4 80 4.5 80-S5 5 100-YF, 2080s 5 5 80 4.5 80-S6 6 500-YF, 2080s 6 6 80 4.5 SD – – – – – –

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conditioned on right-time and right-level investments. The investment decisions are however made through four sets of auxiliary networks each represents a type of RES. Within each of the auxiliary network, three sub-decisions are to be determined: (1) whether the investment should be made, (2) what level of investment should be made (assuming 6 levels each linked to a severity expected to be imposed by a scenario), and (3) the optimum timing in case of investment.

If the model selects no investment (meaning the element e remains vulnerable) in any of the 2020, 2050, and 2080, the unit flow supplied by SS node passes the transshipment node “No Inv.”. Otherwise, the flow passes through one of the six investing-nodes meaning one RES is considered to protect the element’s operability. While the investment is made, the model doesn’t allow the element to operate in the performance lower than what assigned to the invested strategy. Also, we assume the element e remains vulnerable if investment level doesn’t match the severity of the flooding event assigned to the scenario.

Constraint (26) shows the optimum decision model makes for three sub-decisions discussed above. Considering all arcs (i, j) connecting transshipment nodes, this scenario-specific constraint seeks the one carries flow. The loaded arc corresponds to a severity level (i.e. one of the rows in Fig. B1) and investing time (i.e. one of the columns in Fig. B1). The former, severity level, is determined by a binary variable denoted as − +j s ε

6 . Since the investment cannot be made on levels weaker than scenario s’s required RES, the

successful investment is made in scenario s’s corresponding level or stronger. E.g. if scenario s and investment level are equal, the expression holds value 1, otherwise 0.

The latter, investment time, is determined by a binary variable denoted as + ′s j 6

. This expression matches the time in which RES level j is assigned to be invested in and the scenario s’s expected time. E.g. if investment occurs in year 2050 (meaning ord(j′) holds value 2.5), during the 2020 extremes, such investment cannot enhance the resilience of the element since the expression above holds value 0. Constraint (27), (28) assure the unit flow is sent out from and sunk at SS and SD nodes, respectively. Constraint (29) represents flow conservation at transshipment arcs and constraints (30), (31) assure the binary variable IVNr e

i j , , holds value 1, if the

corresponding arc carries the flow. The following represents the constraint (26) syntax in GAMS.constraint_name (r, s, e).

IVF_e(r, s, e) = e = sum((i, j), IVN(r, e, i, j) ∗ (ceil((f(j)-f_doubleprime(s) + eps)/(six))) ∗ (ceil((f_doubleprime(s)-f_prime(j)/ (six))));

B.2. Constraint (33)

The auxiliary network and input parameters for constraint (33) are as follows (see Fig. B2). The ord(s) for investment plans for 2020, 2050, and 2080 is 1.5, 3.5, and 5.5., respectively.

Appendix C. Supplementary material

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.tre.2018.02. 009.

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Fig. B2. Auxiliary network for tanker’s asset allocation graph.

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