Simultaneous optimization of evacuation route and departure time based on link-congestion mitigation

Abstract

Link-level congestion, such as link spillback and long intersection queues, should be avoided during emergency evacuation. The reason is that these local traffic incidents can cause traffic safety risks and hinder evacuation tasks. To determine reliable routes and departure times for the whole evacuation, we establish the link-congestion mitigation-based dynamic evacuation route planning (LCM-DERP) model. The distinct difference with the typical DERP model lies in the objective composition. The system cost objective in our model consists of not only total evacuation time but also external congestion cost. The penalization for link spillback and long intersection queues is used to represent external congestion cost. An improved cell transmission structure, composed of tail cell and head cell and approach cell, is proposed to simulate dynamic traffic flow. Specifically, tail cell and head cell can detect the information of link spillback and long intersection queue separately. This function enables the representation of external congestion cost expressed by multiplying link-level congestion vehicles with penalty parameter. A method of successive average, including a calculation of the local path marginal cost, is used to solve the model. We applied the LCM-DERP model on a road network around Olympic Stadium in Nanjing, China, to test its effectiveness in the aspect of link-congestion control. Compared with the typical DERP model, our method can improve system cost, especially in the high demand range, wherein the reduced external congestion cost is larger than this reduced system cost.

Appendices

Appendix A. Notations

Symbol definition related to cell structure

Symbol	Description
T	Set of loading time interval t
I	Set of all the cells
I − i	The upstream cell set of cell i
I + i	The downstream cell set of cell i
Q t i	The capacity of cell i at time interval t
N t i	The maximum number of vehicles that can be presented in cell i at time interval t
L i	The number of unit cells in cell i
n t i	The vehicle occupancy of cell i at time interval t
y t i,j	The flow from cell i to j at time interval t
y t i	The outflow from cell i at time interval t, expressed as $y_{i}^{t} = \sum\nolimits_{{k \in I_{i}^{ + } }} {y_{i,k}^{t} }$
S t i	The sending flow from cell i at time interval t
R t i	The receiving flow to cell i at time interval t
β t i	The vehicle occupancy heading for branch cell i divided by the one heading for all the branch cells from upstream neighboring diverge cell at time interval t
γ t i	The ratio of vehicle occupancy in cell i to the one in all the parallel merge cells at time interval t

Symbol definition related to model

Symbol	Description
f rs p,τ	Flow from OD pair rs entering route p at departure time interval τ
f	Path flow vector, including the information of departure time and passing routes, whose integration in the network can be presented as path flow pattern f
F	Set of dynamic path flow
RS	Set of OD pairs
P rs	Set of routes connecting OD pair rs
T d	Set of departure time interval τ in the whole demand departure horizon
$\phi_{p,\tau }^{rs} \left( {\mathbf{f}} \right)$	Generalized cost met by evacuees entering path p ∈ P rs at departure time interval τ, which is a unique mapping with respect to f
q rs τ	Demand between OD pair rs at departure time interval τ
Q rs	Total demand for OD pair rs during the study time horizon
η	Penalty coefficient for the vehicle arrival time at destination
$c_{p,\tau }^{rs} \left( {\mathbf{f}} \right)$	actual path travel time for flow entering path p ∈ P rs at departure time interval τ, which is a unique mapping with respect to f
M i	Unit penalty value for the vehicle who confronts traffic congestion in cell i during certain loading time interval
I ′	Set of head cells and tail cells
φ i,t	Judgment coefficient regarding the happening of traffic congestion in cell i at time interval t

Appendix B: Specific Local PMCs Departing from Cells of Different Types

2.1 PMC departing from tail cell

Given an increased vehicle in tail cell i toward downstream head cell j at time interval t, the PMC relation would be depicted in Formula (B.1).

$${\text{PMC}}_{p,t}^{is} = {\text{PMC}}_{{p^{\prime},t + t_{i,j} }}^{js} - {\text{CMC}}_{j}^{{p^{\prime},t + t_{i,j} }} + {\text{CMC}}_{j}^{p,t} + {\text{CMC}}_{i}^{p,t}$$

(B.1)

Except for the first term, the result of other components is dependent on the traffic conditions in cell i and j. The second and third terms can be offset in the first two constraints, and thus, these two values do not need to be calculated. Moreover, for easy analysis, the bottleneck here means that the vehicle speed is below the free-flow speed.

• 1. No bottleneck in cell i

Because no delay occurred in cell i, CMC ^p,t _i will be the free-flow time of cell i, which is equal to l _i time intervals.

• 2. Bottleneck in cell i and no bottleneck in cell j

As shown in Figure 7a, CMC ^p,t _i is equal to t ^D _i  − t + M _i  · Γ(t, t ^B _i ). In fact, CMC ^p,t _i is composed of three elements. The first element consists of the travel time of the additional vehicle, t ^B _i  − t. The second element is the increased delay of the following vehicles, t ^D _i  − t ^B _i (deduced from the rule that the area between the arrival curve and departure curve is equal to total travel time). The third element is the congestion penalization fee added to the additional vehicle and is equal to M _i  · Γ(t, t ^B _i ), where Γ(t, t ^B _i ) is the total number of queue spillback times among the horizon from t to t ^B _i . According to model description, when a vertical gap between the departure and arrival curves at certain time interval t ^′ falls within the horizon of [N ^t _i  − Q ^t _i , N ^t _i ], we consider it queue spillback time. As Figure 7a illustrates, if the spillback only occurs between horizons t ^E _i and t ^F _i , we can set their occurring numbers as t ^F _i  − t ^E _i (in terms of time interval units).

Fig. 7

Cumulative flow curve of congested cell. a Upstream cell i. b Downstream cell j

• 3. Bottleneck in cell i and cell j

As shown in Figure 7, because the influence of additional vehicle in cell i can propagate to cell j, these two cells should be imagined as a single virtual cell to calculate the marginal cost. To be convenient, ${\text{CMC}}_{j}^{{p^{\prime},t + t_{i,j} }}$ and CMC ^p,t _i  + CMC ^p,t _j should be figured out separately. ${\text{CMC}}_{j}^{{p^{\prime},t + t_{i,j} }}$ is equal to $t_{j}^{D} - t_{i}^{B} + M_{j} \cdot \Gamma \left( {t_{i}^{B} ,~t_{j}^{B} } \right)$, whose calculation process is akin to the one at the second condition. CMC ^p,t _i  + CMC ^p,t _j is equal to $t_{j}^{D} - ~t + M_{i} \cdot \Gamma \left( {t,~t_{i}^{B} } \right) + M_{j} \cdot \Gamma \left( {t_{i}^{B} ,~t_{j}^{B} } \right)$ which is composed of three elements. The first element is the travel time of the additional vehicle, which can be represented by $t_{j}^{D} - ~t + M_{i} ~ \cdot ~\Gamma \left( {t,~t_{i}^{B} } \right) + M_{j} ~ \cdot ~\Gamma \left( {t_{i}^{B} ,~t_{j}^{B} } \right)$. The second element is the increased delay of the following vehicles, which is denoted by t ^D _j  − t ^B _j . The third element is the penalized congestion fee, which is expressed as M _i  · Γ(t, t ^B _i ) + M _j  · Γ(t ^B _i , t ^B _j ) (Fig. 7).

2.2 PMC departing from approach cell

Given an increased vehicle in approach cell i toward downstream tail cell j at time interval t, the PMC relation would be depicted in Formula (B.2).

$${\text{PMC}}_{p,t}^{is} = {\text{PMC}}_{{p^{\prime},t + t_{i,j} }}^{js} - {\text{CMC}}_{j}^{{p^{\prime},t + t_{i,j} }} + {\text{CMC}}_{j}^{p,t} + \sum\nolimits_{{k \in I_{j}^{ - } }} {{\text{CMC}}_{k}^{p,t} }$$

(B.2)

Regardless of which traffic state occurred in these merge cells, $\sum\nolimits_{{k \in I_{j}^{ - } }} {{\text{CMC}}_{k}^{p,t} }$ will always be identical to CMC ^p,t _i as stated by Qian et al. (2012). Thus, the PMC result from tail cell can still be applied in this situation.

It is noted that we establish all downstream cells of origin cells as approach cells.

2.3 PMC departing from head cell

Given an increased vehicle in head cell i toward downstream approach cell j at time interval t, the PMC relation would be depicted in Formula (B.3).

$${\text{PMC}}_{p,t}^{is} = {\text{PMC}}_{{p^{\prime},t + t_{i,j} }}^{js} - {\text{CMC}}_{j}^{{p^{\prime},t + t_{i,j} }} + {\text{CMC}}_{i}^{p,t} + \sum\nolimits_{{k \in I_{i}^{ + } }} {{\text{CMC}}_{k}^{p,t} }$$

(B.3)

If cell i is not a bottleneck, the results of PMC departing from the tail cell can still be employed regardless of whether cell j is congested. Thus, we only discuss the condition in which a bottleneck occurs in cell i (in addition, assume at least two downstream cells exist). In this condition, ${\text{CMC}}_{j}^{{p^{\prime},t + t_{i,j} }}$ is t ^D _j  − t ^B _i  + M _j  · Γ(t ^B _i , t ^B _j ), the same with the result for sequential cells. ${\text{CMC}}_{i}^{p,t} + \sum\nolimits_{{k \in I_{i}^{ + } }} {{\text{CMC}}_{k}^{p,t} }$ becomes two elements of composed cost. The first element is the usual increased cost, which is represented by $t_{j}^{D} - t~ + M_{i} ~ \cdot ~\Gamma \left( {t,~t_{i}^{B} } \right)~ + M_{j} \cdot ~\Gamma \left( {t_{i}^{B} ,~t_{j}^{B} } \right)$. The second element is the unusual additional delay, which is named jump-point caused delay in Qian et al. (2012). The embedding of this delay is to release an unreasonable assumption: The flow in cell k is not influenced by the flows at i and j. In fact, the influence exists. It is caused by the rounding calculation of Formula (3) combined with the vehicle sequence information. Its explanation and calculation process are detailed in Zhengfeng et al. (2014). We only show the unusual delay calculation formula here to keep the marginal cost intact: $\sum\nolimits_{{t^{\prime} \in \left[ {t_{i}^{B} ,t_{i}^{D} } \right]}} {\sum\nolimits_{{k \in I_{i}^{ + } }} {\theta_{{t^{\prime}}}^{k} \cdot \delta_{{t^{\prime}}}^{k} } }$. Congestion state variable $\theta_{{t^{\prime}}}^{k}$ represents the congestion state of cell k at time interval t′; 1 signifies congestion and −1 signifies no congestion. Perturbation variable $\delta_{{t^{\prime}}}^{k}$ represents change in terms of cumulative vehicles of cell k at time interval t′, when referring to the original simulated traffic. The assigned number 1 (0 or −1) represents 1 vehicle exceeding (no change compared to or 1 vehicle less than) the original cumulative vehicles.

2.4 PMC departing from origin cell

Our method can only attain the local PMC in terms of loading time interval t. However, the local PMC in terms of departure time interval is required in the MSA algorithm. Because the time span of the departure time interval τ is longer than the loading time interval, so we need to construct a relation between these two types of local PMC to obtain the function of the one from the departure time interval. In the subsequent part, first we provide the formula of PMC ^is _p,t , where t∈τ; then we describe the corresponding PMC ^is _p,τ .

For easy calculation, we make three assumptions. (1) The added vehicle will embed into the end of demand queue at each departure time interval. (2) The influence of the added vehicle to other vehicles in the origin cell of the following departure time intervals can be neglected. (3) The influence to the downstream neighboring cell can be neglected. The local PMC of added vehicle departing from the origin cell i at t can be simplified as Formula (B.4).

$${\text{PMC}}_{p,t}^{is} = {\text{PMC}}_{{p^{\prime},t^{ \hbox{min} } + t_{i,j}^{\hbox{min} } }}^{js} + {\text{CMC}}_{i}^{{p,t^{\hbox{min} } }} , \quad t \in [t^{\hbox{min} } ,t^{\hbox{max} } )$$

(B.4)

where t ^min and t ^max are the lower and upper loading time intervals within the departing time interval τ. t ^min _i,j is the time span from the beginning of τ to the time in which all demands in i from τ are cleaned up, and this value is also assigned to ${\text{CMC}}_{i}^{{p,t^{\hbox{min} } }}$. The PMC of added vehicle departing in the next cell of the subsequent time interval is expressed by the first term of right side. Formula (B.4) reveals that the PMCs of vehicles leaving within the same departure time interval are equal. Their values could be the PMC of vehicles departing at the corresponding last loading time interval. This feature is useful to the work of time interval conversion. It denotes that the common value PMC ^is _p,t for each loading time interval t within the same departure time interval τ can be assigned to PMC ^is _p,τ .