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Recent studies have observed hysteresis loops in the macroscopic fundamental diagram (MFD). In particular, for the same network density, higher network flows occur during congestion onset than during congestion offset. To evaluate management strategies using the MFD, investigating the relationship between the size of these loops and network performance is needed. The existing literature has mainly discussed correlating loop width (difference in density) and height (capacity drop) with congestion heterogeneity, but has failed to prove a relationship between the capacity drop and traffic conditions. Moreover, quantification of the MFD loop in complex multimodal networks has not been investigated. The objective of this paper covers these aspects. We simulated the Sioux Falls network with different mode-share ratios (car and bus users) based on a multi-agent simulation, MATSim. We investigated the relationships between MFD loop size and congestion heterogeneity (standard deviation of density) and network performance (average passenger travel time), and found that both were directly correlated with loop width, while weakly correlated with loop height. Moreover, we divided the MFD loop into two parts according to congestion onset and offset periods and found that the heights of the two parts had opposite effects. Accordingly, we show why the relationship between capacity drop and congestion heterogeneity is not found in the literature. We also found that network performance inversely affected the height of part of the loop while the height of its other part increased with an increase in congestion heterogeneity. These results help to evaluate network performance in the presence of MFD hysteresis, leading to elaborated management decisions.

Traffic congestion is a severe matter worldwide as it causes delays and air pollution. Thus, several traffic-management strategies have been developed to control traffic congestion. However, evaluating the efficiency of any traffic management strategy prior to its implementation is a challenge due to the wide variety of control variables that make the joint optimization of all traffic-flow measures difficult. This is, for instance, due to the large data sets that need to be processed and the large solution space needed. Alternatively, the macroscopic fundamental diagram (MFD), proposed by [

Buisson and Ladier [^{ } [

Recently, Saberi and Mahmassani [

The next section reviews the relevant literature with an overview of using MFDs as an indicator of network performance under different traffic control strategies. Section 3 illustrates the simulation tool used in this paper and the development of MFDs for different network characteristics and travel choices. Section 4 presents the results of the proposed quantification of the MFD hysteresis loop and its relationships. Section 5, finally, presents a summary and conclusions.

The MFD, developed earlier by [

MFDs have been used to represent traffic evolution and to evaluate network performance. Geroliminis and Daganzo [

MFDs have been used to evaluate the performance of transport networks subject to a zone-based routing system and cordon pricing (e.g., [

Other researchers (e.g., [

In summary, analyses using MFDs for evaluating the effects of different traffic management strategies suggest that networks can be evaluated based on this approach. According to changes in critical density and maximum flow values, the performance of networks can be qualified under different traffic management strategies. Additionally, the size of the drop in MFD hysteresis loops when congestion resolves may reflect congestion severity.

Early investigation of hysteresis phenomena belongs to [

・ Differences between highways, surface streets, city center, and suburban roads.

・ When demand is not distributed uniformly, congestion may occur in some parts of the network.

・ The appearance and disappearance of congestion can also be a reason for successive values of mean flow.

・ Differences in data-measurement locations: some loops are located near traffic signals, and some are much farther away.

For urban networks, Mazloumian et al. [^{ } [

In response to the appearance of MFD hysteresis loops, some researchers had developed clustering algorithms to divide the network into homogeneous sub-networks to get low scattered MFD (e.g., [

Saberi and Mahmassani [

Although these studies ( [

Orfanou et al. [

Like MFD hysteresis loop, [

In summary, this review shows that characterizing MFD hysteresis loops and their relationships to traffic characteristics is still an ongoing research topic. Few methodologies have been employed and different assumptions have been made regarding the formation of hysteresis loops and the factors that influence them. Little attention has been paid, however, to quantifying MFD hysteresis loops and their relation with congestion inhomogeneity, network performance, and other traffic characteristics. Those who have attempted to do so have used limited sample sizes and studied specific loop shapes and have found, in the main, relationships between loop size and heterogeneity, not network performance. Additionally, the literature has provided no relation between the capacity drop on hysteresis loops and traffic conditions. Moreover, quantifying hysteresis loops in complex multimodal networks has not been investigated. Closing these gaps in the literature is the objective of this paper.

Vehicle and passenger travel behavior can be estimated based on the MATSim framework, which uses an iterative approach for agent-based dynamic traffic assignment. Integrated simulation of private and public transport based on a queuing model allows time-dependent calculation of travel times, accounting for spillover effects and the direct interaction of private and public transport. Agents alter their behavior from iteration to iteration, based on a co-evolutionary algorithm to try to find optimal routes, modes, and departure times and, thus, maximize the total utility of their daily activity schedule. Following each iteration of the queue-based network assignment, the choices of each agent are evaluated and scored, allowing agents to select more successful options for execution of their schedule in the next iteration. The selection of travel alternatives from the choice set of each agent is performed based on a random utility model, which, after a number of iterations, leads to the convergence of individual and total utilities and, thus, to an agent-based SUE [

The extended Sioux Falls network [

We tested different supply characteristics and different mode-choice combinations to consider a variety of MFD loops. The supply characteristics of scenarios 1, 2, and 3, shown in

Scenario | Network characteristics | Mode-share ratio | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Link type | No. of lanes | Free flow speed km/h | Flow capacity veh/h | Car % | Bus % | walk% | ||||

Urban | Highway | Urban | Highway | Urban | Highway | |||||

Scenario 1 | Urban | 2 | ------ | 50 | ------ | 2000 | ------ | 78% | 19% | 3% |

Scenario 2 | Urban | 2 | ------ | 50 | ------ | 1500 | ------ | 78% | 19% | 3% |

Scenario 3 | Urban | 1 | ------ | 50 | ------ | 2000 | ------ | 78% | 19% | 3% |

Scenario 4 | Urban + Highway | 2 | 3 | 50 | 90 | 800 - 1000 | 1700 - 1900 | 58% | 34% | 8% |

Scenario 5 | Urban + Highway | 2 | 3 | 50 | 90 | 800 - 1000 | 1700 - 1900 | 55% | 36% | 8% |

Scenario 6 | Urban + Highway | 2 | 3 | 50 | 90 | 800 - 1000 | 1700 - 1900 | 52% | 39% | 9% |

12:00 a.m. to 11:59 p.m. Mode-choice rate was controlled by changing the disutility for traveling by car. The marginal disutility for the bus mode was set by referring to [

Q t = ∑ i ∈ z q i , t l i ∑ i ∈ z l i (1)

K t = ∑ i ∈ z k i , t l i ∑ i ∈ z l i (2)

σ t = ∑ i ∈ z ( l i ( k i , t − K ¯ t ) 2 ) ∑ i ∈ z l i (3)

σ ¯ h y s = ∑ t ∈ h y s ∑ i ∈ z ( l i ( k i , t − K ¯ t ) 2 ) ∑ i ∈ z l i (4)

T T t = ∑ j ∈ D T t T j (5)

T T ¯ h y s = ∑ t ∈ h y s ∑ j ∈ D T t T j ∑ j ∈ D T t D T t (6)

where,

Q t : Weighted average of network flow at time t (veh/h).

K t : Weighted average of network density at time t (veh/km).

q i , t : Flow of link i at time t.

k i , t : Density of link i at time t.

l i : Length of link i.

z : Set of links in the network.

k ¯ t : Average of link densities at time t.

σ t : Standard deviation of density at time t.

σ ¯ h y s : Average of standard deviation of density over a hysteresis loop.

T j : Trip travel time of traveler j (min).

D T t : Set of traveler whose departure time is t.

T T ¯ h y s : Average passenger travel time over a hysteresis loop.

Based on [

Previous research approximated the size of a hysteresis loop as the product of its width and height. To ensure the accuracy of this approximation, we calculated the actual size of the loop for a random two-thirds sample of MFDs shown in ^{2}) value found between the actual area and its approximation, when correlated with the standard deviation of density, the approximated area size

showed almost the same correlation trend and accuracy as with the actual one. Thus, for simplicity, we determined the next set of relationships using the approximated area size (by multiplying the width and height of the loop). Different relationships were found between the average standard deviation of density (for the recovery phase only and considering both the loading and recovery phases) and loop metrics. ^{2} values for those relationships.

Whole loop | Part 1 loop | Part 2 loop | |||||||
---|---|---|---|---|---|---|---|---|---|

(∆q*∆k) | ∆k | ∆q | (∆q_{1}*∆k_{1}) | ∆k_{1} | ∆q_{1} | (∆q_{2}*∆k) | ∆k | ∆q_{2} | |

Average standard deviation of density (calculated over recovery phase only) | 0.762 | 0.634 | 0.046 | 0.554 | 0.570 | −0.320 | 0.862 | 0.634 | 0.841 |

Average standard deviation of density (calculated over loading and recovery phases) | 0.784 | 0.721 | 0.021 | 0.578 | 0.646 | −0.390 | 0.902 | 0.721 | 0.857 |

where

Max point: point at maximum flow.

Start point: point at loop start.

End point: point at loop ends.

Whole loop: loop with height ∆q and width ∆k.

Part 1 loop: congested part with height ∆q_{1} and width ∆k_{1}.

Part 2 loop: recovery part with height ∆q_{2} and width ∆k.

Judging from ^{2} value when the average standard deviation of density was calculated over the recovery period only, was 0.762. However, when the average standard deviation of density was calculated over loading and recovery periods, the R^{2} value was 0.784. Additionally, a weak correlation was found between loop height and standard deviation of density (as in the results of [

・ The correlation between the standard deviation of density and the metrics of the Part 2 loop are stronger than for the overall loop. That is, the Part 2 loop alone can represent the heterogeneity effect.

・ The correlation between the standard deviation of density and the heights of two loop parts (R^{2} = 0.390 and 0.857 for ∆q_{1} and ∆q_{2}, respectively) are stronger than its correlation with overall loop height (R^{2} = 0.021). Additionally, the two heights are oppositely correlated to the standard deviation of density (

1) ∆q_{1} is the capacity drop that occurred while demand was still high and the network was loading, and is associated with the inability of the network to sustain its throughput at its maximum value for a long time [

this increased congestion causes a decrease in network flow and, consequently, an increase in ∆q_{1}. Thus, ∆q_{1} increases with the decrease in standard deviation of density.

2) ∆q_{2} occurred during the recovery process. During congestion dissipation, the densities of many links will decrease; however, some links are still in congestion, leading to a higher standard deviation of density among links. The flow on most links will decrease (due to congestion or due to low link occupancy) and, consequently, ∆q_{2} will increase. Thus, the larger is ∆q_{2}, the more heterogeneous is the congestion distribution during the recovery phase.

To compare a wide range of mode-share ratios and, consequently, a variety of network performance conditions, the scenarios generated by [

Car disutility (βc) | ||||||
---|---|---|---|---|---|---|

Base case | 1 | 0.5 | 0 | −0.5 | −1 | |

Bus % | 19% | 29% | 30% | 34% | 36% | 39% |

Car % | 78% | 65% | 62% | 58% | 55% | 52% |

Walking % | 3% | 6% | 7% | 8% | 8% | 9% |

performance (measured by the average passenger travel time) was tested.

^{2} values for the relationships between the average passenger travel time and loop metrics. A direct correlation was found between the average passenger travel time and loop size and width. Also, a weak relationship was found between the average passenger travel time and overall loop height, which is the same result as the relationship between the standard deviation of density and whole loop height (^{2} = 0.701). However, a weak correlation was found between the average passenger travel time and the height of Part 2 of the loop (R^{2} = 0.510) (

Whole loop | Part 1 loop | Part 2 loop | |||||||
---|---|---|---|---|---|---|---|---|---|

(∆q*∆k) | ∆k | ∆q | (∆q_{1}*∆k_{1}) | ∆k_{1} | ∆q_{1} | (∆q_{2}*∆k) | ∆k | ∆q_{2} | |

Average passenger travel time | 0.860 | 0.883 | 0.336 | 0.767 | 0.806 | −0.701 | 0.888 | 0.883 | 0.510 |

flow correlated with an increase in ∆q_{1}. While ∆q_{2} occurs during congestion dissipation, as some link densities decrease and some links are still in congestion, this leads to a higher standard deviation of density among links that is correlated with the ∆q_{2} increase.

From _{1}. Consequently, a higher average passenger travel time occurs. However, ∆q_{1} decreases according to its inverse relationship to average passenger travel time; this is accompanied by a decrease in traffic flow and a large increase in density (large ∆k_{1}). This increase in density can be eliminated by reducing the number of car users. The scenario of “58% Cars” results in light congestion, indicated by a decrease in traffic flow and a small increase in density, accompanied by an increase in ∆q_{1} and, consequently, a lower average passenger travel time. An additional decrease in the percentage of car users, while keeping the bus fleet the same, would allow the network to operate with nearly free flow: see the results for the “52% Cars” and “55% Cars” modes.

MFDs that relate the average network density and flow have been used in the literature for network performance evaluation. However, it is difficult to attain low scattered MFDs in real-world city-scale networks: instead, recent studies have suggested more complex MFDs that exhibit a hysteresis loop. In particular, for the same network density, a higher network flow occurs during congestion onset than during congestion offset. Investigating the relationship between these loop sizes and network performance is of great importance in comparing the effect of different management strategies. This study investigated quantifying MFD hysteresis loops and the relationship between the size of this loop and both congestion heterogeneity and network performance in multimodal networks.

We estimated MFDs for traffic conditions associated with different mode-share ratios (car and bus users) on the Sioux Falls network, based on a multi-agent transport simulation. Mode choice was based on a random utility model with an agent-based stochastic user equilibrium. The goal was to compare traffic conditions associated with different mode-share ratios, based on MFDs, and, consequently, evaluate the effect of different management strategies on network performance in the presence of a hysteresis loop. First, we evaluated a quantification method for MFD loops. Previous research [

By correlating with the standard deviation of density, the approximated area showed almost the same correlation trend and accuracy as did the actual area. We conclude that a hysteresis loop’s width (difference in density), its height (capacity drop), and its area (by multiplying the loop’s width and height) are representative metrics for MFD loops. Different relationships were found between the average standard deviation of density and loop metrics. It is concluded that the average standard deviation of density should be calculated over both loading and recovery periods, and not for the recovery period alone, especially for loops covering a long period. Dividing loop area into two parts according to congestion onset and offset times (upper and lower parts, respectively) provides some important insights. We consider, to some extent, that the partitioning criteria for a loop’s area suggested in this paper are helpful for understanding a number of relationships that have not been found in the literature. It was found that the correlation between the standard deviation of density and the loop’s lower part is stronger than its correlation with the overall loop. That is, the lower part of the loop alone can represent the heterogeneity effect. We also found that the correlation between the standard deviation of density and the heights of both loops’ parts was stronger than the correlation with overall loop height. This is because the two heights have opposite effects on the standard deviation of density, which affects the correlation with the overall loop height. This illustrates why no relationship between capacity drop and congestion inhomogeneity has been found in previous research ( [

For network performance, a clear inverse relationship was found between the average passenger travel time and the height of the upper part of the loop (R^{2} = 0.70), while it was weakly correlated with the height of the lower part of the loop (R^{2} = 0.51). Additionally, useful conclusions were reached when comparing the correlation between the metrics of the MFD loop suggested in this paper and both the standard deviation of density and the average passenger travel time. It was found that the standard deviation of density was strongly and directly correlated with the height of the lower part of the loop while the average passenger travel time had a strong inverse correlation with the height of the upper part of the loop. We conclude that the height of the upper part of the loop is sensitive to the average passenger travel time whereas the height of the lower part of the loop is more sensitive to the standard deviation of density. That is, network performance inversely affects the capacity drop while the network is loading, whereas the inhomogeneous spatial distribution of congestion increases the flow reduction during network unloading with congestion dissipation. These findings may help in using the hysteric MFDs that are expected for city-scale networks. This may provide a simple tool for comparing the effects of management strategies on network performance, and contribute to more elaborate decisions. Further statistical analysis would benefit from a larger sample of hysteric MFDs to test properly for the representational power and generalization ability of the developed relationships.

Hemdan, S., Wahaballa, A.M. and Kurauchi, F. (2018) Quantification of the Hysteresis of Macroscopic Fundamental Diagrams and Its Relationship with the Congestion Heterogeneity and Performance of a Multimodal Network. Journal of Transportation Technologies, 8, 44-64. https://doi.org/10.4236/jtts.2018.81003