Blouin in the wind 311 Blowin' in the wind Mark Wagner Kenneth Kot William e kolasa Lawrence Technological University Southfield. MI Advisor: Ruth G. favro Introduction We present a model to determine the optimal evacuation plan for coastal South Carolina in the event of a large hurricane. The model simulates the flow of traffic on major roads. We explored several possible evacuation plans, comparing the time each requires Traffic flow can be significantly improved by reversing the eastbound lanes of 1-26 from Charleston to Columbia. By closing the interchange between I-26 and I-95 and restricting access to I-26 at Charleston, we can reduce the overall evacuation time from an original 31 h to 13 h. However, astaggered evacuation plan, which evacuates the coastline county by county, does not improve the evacuation time, since traffic from each coastal population center interferes little with traffic flowing from other areas being evacuated. Although reversing traffic on other highways could slightly im prove traffic flow, it would be impractical. Restrictions on the number and types of vehicles could speed up the evacuation but would likely cause more problems than improvements Theory of Traffic Flow We require a model that simulates traffic flow on a large scale rather than individual car movement. We take formulas to model traffic flow from beltrami [1998]. Although traffic is not evenly distributed along a segment of road, it can be modeled as if it were when large segments of road are being considered We can measure the traffic density of a section of road in cars/ mi. The traffic The UMAP Journal 22(3)(2001)311-321. Copyright 2001 by COMAP, Inc. All rights reserved ermission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
Blowiní in the Wind 311 Blowiní in the Wind Mark Wagner Kenneth Kopp William E. Kolasa Lawrence Technological University Southfield, MI Advisor: Ruth G. Favro Introduction We present a model to determine the optimal evacuation plan for coastal South Carolina in the event of a large hurricane. The model simulates the flow of traffic on major roads. We explored several possible evacuation plans, comparing the time each requires. Traffic flow can be significantly improved by reversing the eastbound lanes of I-26 from Charleston to Columbia. By closing the interchange between I-26 and I-95 and restricting access to I-26 at Charleston, we can reduce the overall evacuation time from an original 31 h to 13 h. However, a staggered evacuation plan, which evacuates the coastline county by county, does not improve the evacuation time, since traffic from each coastal population center interferes little with traffic flowing from other areas being evacuated. Although reversing traffic on other highways could slightly improve traffic flow, it would be impractical. Restrictions on the number and types of vehicles could speed up the evacuation but would likely cause more problems than improvements. Theory of Traffic Flow We require a model that simulates traffic flow on a large scale rather than individual car movement. We take formulas to model traffic flow from Beltrami [1998]. Although traffic is not evenly distributed along a segment of road, it can be modeled as if it were when large segments of road are being considered. We can measure the traffic density of a section of road in cars/mi. The traffic The UMAP Journal 22 (3) (2001) 311ñ321. c Copyright 2001 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
312 The umaP Journal 22.3(2001) speed u at a point on the road can be calculated from the density according to the formula u(r)=um(1-p where p is the traffic density, um is the maximum speed of any car on the road, and pm is the maximum traffic density(with no space between cars). We define the flow of traffic at a point on the road as the number of cars passing that point in a unit of time. The flow q can be easily calculated as qp)=p It is the flow of traffic that we desire to optimize, since greater flow results in a greater volume of traffic moving along a road Assumptions During an evacuation, there is an average of 3 people per car. This is rea sonable, since people evacuate with their entire families, and the average household in South Carolina has 2.7 people, according to the 1990 census The average length of a car on the road is about 16 ft In a traffic jam there is an average of 1 ft of space between cars The two above assumptions lead to a maximum traffic density of 5280 ft/mile 310 cars/mile/lane 17 ft/car The maximum speed is 60 mph on a 4-lane divided highway, 50 mph on a 2-lane undivided country road Vehicles follow natural human tendencies in choosing directions at intersec tions, such as preferring larger highways and direct routes The traffic flow of evacuees from Florida and Georgia on I-95 is a continuous stream inward to South carolina When vehicles leave the area of the model, they are considered safely evac- uated and no longer need to be tracked There will not be traffic backups on the interstates at the points at which they leave the area of the model A maximum of 30 cars/min can enter or exit a l-mi stretch of road in a pop ulated area, by means of ramps or other access roads. Up to the maximum exit rate, all cars desiring to exit a highway successfully exit
312 The UMAP Journal 22.3 (2001) speed u at a point on the road can be calculated from the density according to the formula u(r) = um 1 − ρ ρm , where ρ is the traffic density, um is the maximum speed of any car on the road, and ρm is the maximum traffic density (with no space between cars). We define the flow of traffic at a point on the road as the number of cars passing that point in a unit of time. The flow q can be easily calculated as q(ρ) = ρu. It is the flow of traffic that we desire to optimize, since greater flow results in a greater volume of traffic moving along a road. Assumptions • During an evacuation, there is an average of 3 people per car. This is reasonable, since people evacuate with their entire families, and the average household in South Carolina has 2.7 people, according to the 1990 census. • The average length of a car on the road is about 16 ft. • In a traffic jam, there is an average of 1 ft of space between cars. • The two above assumptions lead to a maximum traffic density of 5280 ft/mile 17 ft/car = 310 cars/mile/lane. • The maximum speed is 60 mph on a 4-lane divided highway, 50 mph on a 2-lane undivided country road. • Vehicles follow natural human tendencies in choosing directions at intersections, such as preferring larger highways and direct routes. • The traffic flow of evacuees from Florida and Georgia on I-95 is a continuous stream inward to South Carolina. • When vehicles leave the area of the model, they are considered safely evacuated and no longer need to be tracked. • There will not be traffic backups on the interstates at the points at which they leave the area of the model. • A maximum of 30 cars/min can enter or exit a 1-mi stretch of road in a populated area, by means of ramps or other access roads. Up to the maximum exit rate, all cars desiring to exit a highway successfully exit
Blowoin' in the wind 313 The weather does not affect traffic speeds. The justifications are During the early part of the evacuation, when the hurricane is far from the coast, there is no weather to interfere with traffic flowing at the maximum speed possible During the later part of the evacuation, when the hurricane is approach ing the coast, traffic flows sufficiently slowly that storm weather would not further reduce the speed of traffic There is sufficient personnel available for any reasonable tasks Objective statement We measure the success of an evacuation plan by its ability to evacuate all lives from the endangered areas to safe areas between announcement of mandatory evacuation and landfall of the hurricane; the best evacuation plan takes the shortest time Model Design The Traffic Simulator Our traffic simulator is based on the formulas above. Both space and time are discretized, so that the roads are divided into 1-mi segments and time is divided into 1-min intervals. Vehicles enter roads at on-ramps in populated areas, leave them by off-ramps, and travel through intersections to other roads Each 1-mi road segment has a density (the number of cars on that segment), a speed(mph), and a flow( the maximum number of cars that move to the next 1-mile segment in 1 min). Each complete road section has a theoretical maximum density Pm and a practical maximum density Pim(accounting for 1 ft of space between cars), which can never be exceeded Moving traffic along a single road The flow for each road segment is calculated as 9(p)=m If the following road segment is unable to accommodate this many cars, the flow is the maximum number of cars that can move to the next segment
Blowiní in the Wind 313 • The weather does not affect traffic speeds. The justifications are: ñ During the early part of the evacuation, when the hurricane is far from the coast, there is no weather to interfere with traffic flowing at the maximum speed possible. ñ During the later part of the evacuation, when the hurricane is approaching the coast, traffic flows sufficiently slowly that storm weather would not further reduce the speed of traffic. • There is sufficient personnel available for any reasonable tasks. Objective Statement We measure the success of an evacuation plan by its ability to evacuate all lives from the endangered areas to safe areas between announcement of mandatory evacuation and landfall of the hurricane; the best evacuation plan takes the shortest time. Model Design The Traffic Simulator Our traffic simulator is based on the formulas above. Both space and time are discretized, so that the roads are divided into 1-mi segments and time is divided into 1-min intervals. Vehicles enter roads at on-ramps in populated areas, leave them by off-ramps, and travel through intersections to other roads. Each 1-mi road segment has a density (the number of cars on that segment), a speed (mph), and a flow (the maximum number of cars that move to the next 1-mile segment in 1 min). Each complete road section has a theoretical maximum density ρm and a practical maximum density ρ m (accounting for 1 ft of space between cars), which can never be exceeded. Moving Traffic Along a Single Road The flow for each road segment is calculated as q(ρ) = ρu um . If the following road segment is unable to accommodate this many cars, the flow is the maximum number of cars that can move to the next segment.
314 The umaP Journal 22.3(2001) Moving Traffic Through Intersections When traffic reaches the end of a section of road and arrives at an inte section it must be divided among the exits of the intersection For each inter- section, we make assumptions about percentages of cars taking each direction, based on the known road network, the capacities of the roads, and natural hu- man tendencies. If a road ends at an intersection with no roads leading out(i.e the state border), there is assumed to be no traffic backup; traffic flow continues at the highest rate possible, and the simulation keeps track s number of cars that have left the model Conflicts occur when more cars attempt to enter a road section at an inter- section than that road section can accommodate. Consider a section of road that begins at an intersection. Let gmax =pinm-p= the maximum influx of cars the road can accommodate at the intersection q1,...,n=the flows of cars entering the road at an intersection, and qin=2qi=the total flow of cars attempting to enter the road at the intersec tion If qin>qmax, then we adjust the flow of cars entering the road from its entrance roads as follows 9i= gi qma Therefore, qi is the number of cars entering the road from road i. The flow of traffic allowed in from each road is distributed according to the flow trying to enter from each road. Clearly, 24=gmax Simulating Populated Areas A section of road that passes through a populated area has cars enter and leave by ramps or other access roads. We assume that the maximum flow of raffic for an access ramp is 30 cars/min. We estimate the actual number of cars entering and leaving each road segment based on the population of the area Cars cannot enter a road if its maximum density has been reached. For simplicity, however, we assume that cars desiring to exit always can, up to the maximum flow of 30 cars/min per exit ranP of each populated area changes during the evacuation, so that we can determine the time required. Therefore, we keep track of the population in the areas being evacuated, Columbia, and certain other cities in South Carolina. If all people have been evacuated from an area, no more enter the road system from that area Areas do not have to be evacuated immediately when the simulation starts ach area may be assigned an evacuation delay, during which normal traffic is simulated. Once the delay has passed traffic in the area assumes its evacuation behavior
314 The UMAP Journal 22.3 (2001) Moving Traffic Through Intersections When traffic reaches the end of a section of road and arrives at an intersection, it must be divided among the exits of the intersection. For each intersection, we make assumptions about percentages of cars taking each direction, based on the known road network, the capacities of the roads, and natural human tendencies. If a road ends at an intersection with no roads leading out (i.e., the state border), there is assumed to be no traffic backup; traffic flow simply continues at the highest rate possible, and the simulation keeps track of the number of cars that have left the model. Conflicts occur when more cars attempt to enter a road section at an intersection than that road section can accommodate. Consider a section of road that begins at an intersection. Let: qmax = ρ m − ρ = the maximum influx of cars the road can accommodate at the intersection, q1,... ,qn = the flows of cars entering the road at an intersection, and qin = "qi = the total flow of cars attempting to enter the road at the intersection. If qin > qmax, then we adjust the flow of cars entering the road from its entrance roads as follows: q i = qi qin qmax. Therefore, q i is the number of cars entering the road from road i. The flow of traffic allowed in from each road is distributed according to the flow trying to enter from each road. Clearly, "q i = qmax. Simulating Populated Areas A section of road that passes through a populated area has cars enter and leave by ramps or other access roads. We assume that the maximum flow of traffic for an access ramp is 30 cars/min. We estimate the actual number of cars entering and leaving each road segment based on the population of the area. Cars cannot enter a road if its maximum density has been reached. For simplicity, however, we assume that cars desiring to exit always can, up to the maximum flow of 30 cars/min per exit ramp. We desire to know how the population of each populated area changes during the evacuation, so that we can determine the time required. Therefore, we keep track of the population in the areas being evacuated, Columbia, and certain other cities in South Carolina. If all people have been evacuated from an area, no more enter the road system from that area. Areas do not have to be evacuated immediately when the simulation starts. Each area may be assigned an evacuation delay, during which normal traffic is simulated. Once the delay has passed, traffic in the area assumes its evacuation behavior.
Blouin in the wind 315 Completing an Evacuation The six coastal counties of South Carolina(where Charleston includes the entire Charleston area) and the roads leading inland from these areas must be evacuated. When the population of these areas reaches zero and the average traffic density along the roads is less than 5 cars /mi, the evacuation is complete and the simulation terminates Implementing the model We implemented the model described above in a computer program written in C++. The logic for the main function is as follows: For each road, welet traffic exit, resolve traffic at intersections, move traffic along the rest of the road, and finally let cars enter the road. We loop until the evacuation is complete Traffic flow is considered simultaneous; the traffic flow along every road is determined before traffic densities are updated. However, exits occur first and entrances last to accurately simulate traffic at access ramps Model results Simulating the 1999 Evacuation To simulate the evacuation of 1999, we prepared a simplified map that in- ludes the interstates other the 4-lane divided highways, and some 2-lane un divided roads. We simulated the evacuation of the coastal counties Beaufort Jasper, Colleton, Georgetown, and Horry (including Myrtle Beach-and the Charleston metro area. The inland areas we considered are Columbia, Spar- tanburg, Greenville, Augusta, Florence, and Sumter. In addition, we simulated large amounts of traffic from farther south entering 1-95N from the Savannah area. A map of the entire simulation is shown in Figure 1 The results of running this simulation with conditions similar to those of the actual evacuation produced an evacuation time of 31 h to get everyone farther inland than I-95. This is significantly greater than the actual evacuation time and completely unacceptable. The increase in time can be explained by two features of the actual evacuation that are missing in the simulation Only 64% of the population of Charleston left when the mandatory evacu- ation was announced [cutter and dow 2000; Cutter et al. 2000]; our model assumes that everyone leaves Late in the day, the eastbound lanes of 1-26 were reversed, eliminating the congestion
Blowiní in the Wind 315 Completing an Evacuation The six coastal counties of South Carolina (where Charleston includes the entire Charleston area) and the roads leading inland from these areas must be evacuated. When the population of these areas reaches zero, and the average traffic density along the roads is less than 5 cars/mi, the evacuation is complete and the simulation terminates. Implementing the Model We implemented the model described above in a computer program written in C++. The logic for the main function is as follows: For each road, we let traffic exit, resolve traffic at intersections, move traffic along the rest of the road, and finally let cars enter the road. We loop until the evacuation is complete. Traffic flow is considered simultaneous; the traffic flow along every road is determined before traffic densities are updated. However, exits occur first and entrances last, to accurately simulate traffic at access ramps. Model Results Simulating the 1999 Evacuation To simulate the evacuation of 1999, we prepared a simplified map that includes the interstates, other the 4-lane divided highways, and some 2-lane undivided roads. We simulated the evacuation of the coastal countiesóBeaufort, Jasper, Colleton, Georgetown, and Horry (including Myrtle Beach)óand the Charleston metro area. The inland areas we considered are Columbia, Spartanburg, Greenville, Augusta, Florence, and Sumter. In addition, we simulated large amounts of traffic from farther south entering I-95 N from the Savannah area. A map of the entire simulation is shown in Figure 1. The results of running this simulation with conditions similar to those of the actual evacuation produced an evacuation time of 31 h to get everyone farther inland than I-95. This is significantly greater than the actual evacuation time and completely unacceptable. The increase in time can be explained by two features of the actual evacuation that are missing in the simulation: • Only 64% of the population of Charleston left when the mandatory evacuation was announced [Cutter and Dow 2000; Cutter et al. 2000]; our model assumes that everyone leaves. • Late in the day, the eastbound lanes of I-26 were reversed, eliminating the congestion