# Strengthening the Resiliency of a Coastal Transportation System through Integrated Simulation of Storm Surge, Inundation, and Nonrecurrent Congestion in Northeast Florida

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## Abstract

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## 1. Introduction

_{i}and each arc (road) has a capacity of u

_{ij}, and a set of safe nodes S (destinations), find the optimal routes to safety. Optimality here can be defined in different ways: number of evacuees that reach safety, smallest overall time to safety, average time to safety. In this effort, we aim to maximize the number of people that are safely evacuated to one of the nodes in set S.

## 2. Storm Surge and Inundation Modeling within the MTEVA

_{Ncr}.

_{A}, above (or below) the surrounding topography and become unusable if, during the course of a simulation, the water level at any location on the road exceeds some critical height, H

_{Acr}, above the road. If, at any point of time later the water level retreats, the road becomes usable again. Each bridge has its own elevation relative to the simulation vertical datum (e.g., NAVD88), B

_{A}. If, during the course of a simulation the water reaches the bridge, it is considered “destroyed” and permanently unusable. Additionally, regardless of water level, bridges are also assumed to be impassable during periods of high wind when the wind speed exceeds some critical value, W

_{Acr}. For practical purposes H

_{Ncr}is set to 30 cm (about one foot of flooding) and W

_{Acr}to 45 mph. In reality these number can vary depending on location, but they are in line with actual values used by authorities. Critical values can be customized in the system, but not directly in the user interface.

_{ij}and h

_{ij}. The first represents the time/cost to use arc (i,j) ϵ E, while the latter the cost to reverse it, in order to allow the contraflow. It is easy to see that in order to reverse a road, some actions are necessary; police officers should be employed to control traffic, proper traffic signs should be used, etc. For simplicity, we treat every arc the same way, hence c

_{ij }= h

_{ij}= 1.

## 3. The Transportation Network Assignment (aka Evacuation) Model

#### 3.1. Algorithms and Implementation

**Table 1.**Notation used to describe the transportation network,

**G(V, E)**, of a region and the mathematical model.

Sets | |

V | The set of all nodes (intersections) in the network. |

E | The set of all arcs (roads) in the network |

S | The set of nodes that are considered safe |

Input Parameters | |

u_{ij} | The capacity of arc (i, j) ∈ E. For any two nodes (i, j) ∉ E, u = _{ij}0. |

A binary input parameter that is equal to 0 if node i is destroyed at time t, or 1 otherwise. | |

A binary input parameter that is equal to 0 if arc (i, j) is destroyed at time t, or 1 otherwise. | |

K | The budget of arcs that can be reversed during the evacuation process. |

Variables to be Optimized | |

The demand of node i ∈ V at time t ∈ T. is the initial demand of node i and is given. | |

The flow on arc (i, j) ∈ E at time t ∈ T. | |

y_{ij} | A binary variable that is equal to 1 if arc (i, j) ∈ E is reversed, or 0 otherwise. |

_{ij}without consideration of contraflow. The dual multiplier of this set of constraints represents the increase in the number of evacuees, if the capacity of the arcs were bigger. However, this assumes that the number of evacuees is much bigger (which might not be the case). Now, let λ

_{ij}be the dual multipliers of the relaxed constraint. Selecting the K arcs with the biggest λ

_{ij}∗ u

_{ji}gives a greedy approach on the K arcs that should be reversed.

- Time Static [21]
- ○
- Everyone evacuates simultaneously. Future events (congestion/flooding) not considered.
- ○
- Decomposition into discrete, smaller problems, each considering only one time step.

- Time Dynamic Heuristic (Present Study)
- ○
- Evacuation is phased. Future events (congestion/flooding) are considered.
- ○
- Relaxation involves the selection of the arcs to be reversed.
- ○
- Locates the arcs that would benefit evacuation the most (if equal demand everywhere).
- ○
- Resulting formulation of a dynamic network flow problem is solved using the Augmented Lagrange relaxation approach.

max ∑_{t∈T} ∑_{i∈N\S} ∑_{j∈S} | (1) | |

s.t. = + ∑_{j∈N:(j,i)∈E} − ∑_{j∈N:(j,i)∈E} | ∀i ∈ V, ∀t ∈ T: = 1 | (2) |

≤ u_{ij} + y_{i j} u_{j j} | ∀(i, j) ∈ E, ∀t ∈ T: = 1 | (3) |

∑_{(i, j)∈E} y_{ij} ≤ K | (4) | |

≥ 0, | ∀(i, j) ∈ E, ∀t ∈ T | (5) |

≥ 0, | ∀i ∈ V, ∀t ∈ T | (6) |

y_{i j} ∈ {0,1}, | ∀(i, j) ∈ E | (7) |

_{ij}= 0. Equation (4) is a typical budget constraint that ensures no more than K arcs can be reversed. The reason for that limitation is logistical; reversing a road takes time and needs to be done carefully. Hence, it is realistic to assume a limit on that number. Last, constraints (5)–(7) define the restrictions on the variable values.

_{ij}and x

_{ji}. The proof, which is done by contradiction, can be found in Lemma 1 of [13].

#### 3.2. Computational Results

Network Size (nodes) | Average Optimality Gap (%) | Maximum Optimality Gap (%) | Average Time Decrease (%) | Maximum Time Decrease (%) |
---|---|---|---|---|

20 | 0.28 | 1.27 | 81 | 89 |

100 | 0.41 | 3.89 | 85 | 92 |

500 | 0.80 | 8.02 | 87 | 93 |

1000 | 2.54 | 14.47 | 87 | 94 |

10,000 | 8.11 | 31.12 | 89 | 97 |

Network Size (nodes) | Average Optimality Gap (%) | Maximum Optimality Gap (%) | Average Time Decrease (%) | Maximum Time Decrease (%) |
---|---|---|---|---|

20 | 0.19 | 1.01 | 79 | 85 |

100 | 0.24 | 1.15 | 85 | 89 |

500 | 0.55 | 2.01 | 86 | 90 |

1000 | 1.54 | 3.60 | 87 | 90 |

10,000 | 1.99 | 4.41 | 87 | 91 |

Network (nodes) | Average Optimality Gap (%) | Maximum Optimality Gap (%) | Average Time Decrease (%) | Maximum Time Decrease (%) |
---|---|---|---|---|

Sioux Falls (|V| = 24) | 0.00 | 0.00 | 75 | 90 |

Anaheim (|V| = 416) | 1.08 | 1.17 | 85 | 90 |

Winnipeg (|V| = 1057) | 3.02 | 3.66 | 86 | 92 |

Austin (|V| = 7388) | 6.97 | 7.11 | 86 | 94 |

Philadelphia (|V| = 13,389) | 14.90 | 20.00 | 90 | 97 |

Network (nodes) | Average Optimality Gap (%) | Maximum Optimality Gap (%) | Average Time Decrease (%) | Maximum Time Decrease (%) |
---|---|---|---|---|

Sioux Falls (|V| = 24) | 0.00 | 0.00 | 72 | 86 |

Anaheim (|V| = 416) | 0.00 | 0.00 | 76 | 88 |

Winnipeg (|V| = 1057) | 0.00 | 0.00 | 76 | 88 |

Austin (|V| = 7388) | 0.98 | 2.11 | 77 | 92 |

Philadelphia (|V| = 13389) | 3.25 | 3.89 | 82 | 92 |

#### 3.3. Virtual Appliance Performance

Algorithm | Computational Overhead |
---|---|

Time Static | +4.2% |

Time Dynamic | +4.8% |

Heuristic | +6.2% |

## 4. Demonstration Application to the NE Florida Coast

**Figure 1.**User interface for the simple network MTEVA configuration featuring a synthetic tropical storm making landfall in an idealized domain with a bay.

**Figure 2.**Simulated transportation network response to the synthetic storm making landfall in an idealized domain. The initial configuration of the transportation network is shown on the left while the simulated network assignment and storm surge and inundation as the storm approaches is shown on the right.

Heuristic | Average Optimality Gap (%) | Maximum Optimality Gap (%) | Average Time Decrease (%) | Maximum Time Decrease (%) |
---|---|---|---|---|

Static | 14.51 | 14.51 | 88 | 88 |

Dynamic | 4.33 | 4.33 | 80 | 80 |

**Figure 4.**Interface for the Northeast Florida MTEVA configuration showing the track and intensity of the synthetic storm.

**Figure 6.**Comparison of evacuation routes at different time instances three hours apart. In the second case the evacuees are forced to take Acosta Bridge (right) because of the inaccessibility of the Fuller Warren Bridge (which is otherwise the preferred route, left) due to high wind intensity.

**Figure 7.**Raised water level alters local traffic patterns due to flooding of roads and intersections.

**Figure 8.**Estimated percentage of the number of people in flooded areas unable to evacuate (loss) as a function of evacuation start time.

## 5. Summary and Conclusions

- contains a storm surge and inundation modeling system coupled with a transportation network optimization model capable of simulating lane reversal. The coupled modeling system is then applied to both synthetic and real domains.
- demonstrates and promotes interoperability through its use of a THREDDS Data Server (TDS) for distribution and visualization of results. At the most basic level, users can access the MTEVA through the web-based GUI. However, more advanced users are able to setup and perform simulations using the scheduling interfaces directly (e.g., using the “condor_submit” command).
- is completely configurable, customizable and expandable. Because the tools, scripts, web interfaces, etc. are located within the MTEVA; any individual component can be altered to meet an individual user’s needs. For example, locations of nodes modified, additional network nodes/arcs can be added, or demands and capacities changed.
- provides an educational environment useful for students of coastal science, cyberinfrastructure, and transportation engineering. For example, coastal science students can better understand how storm surge impacts a domain given storm strength, domain shape, etc. Cyberinfrastructure students can focus on the technical details of the MTEVA itself along with its web interfaces, databases and scripting technologies used behind the scenes. Transportation engineering students could investigate how the use of lane reversal can be optimized during a storm event. Finally, transportation practitioners in Northeast Florida could use the MTEVA to investigate how their domain responds to different synthetic tropical storms.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Davis, J.R.; Paramygin, V.A.; Vogiatzis, C.; Sheng, Y.P.; Pardalos, P.M.; Figueiredo, R.J. Strengthening the Resiliency of a Coastal Transportation System through Integrated Simulation of Storm Surge, Inundation, and Nonrecurrent Congestion in Northeast Florida. *J. Mar. Sci. Eng.* **2014**, *2*, 287-305.
https://doi.org/10.3390/jmse2020287

**AMA Style**

Davis JR, Paramygin VA, Vogiatzis C, Sheng YP, Pardalos PM, Figueiredo RJ. Strengthening the Resiliency of a Coastal Transportation System through Integrated Simulation of Storm Surge, Inundation, and Nonrecurrent Congestion in Northeast Florida. *Journal of Marine Science and Engineering*. 2014; 2(2):287-305.
https://doi.org/10.3390/jmse2020287

**Chicago/Turabian Style**

Davis, Justin R., Vladimir A. Paramygin, Chrysafis Vogiatzis, Y. Peter Sheng, Panos M. Pardalos, and Renato J. Figueiredo. 2014. "Strengthening the Resiliency of a Coastal Transportation System through Integrated Simulation of Storm Surge, Inundation, and Nonrecurrent Congestion in Northeast Florida" *Journal of Marine Science and Engineering* 2, no. 2: 287-305.
https://doi.org/10.3390/jmse2020287