Content uploaded by Vikash Gayah
Author content
All content in this area was uploaded by Vikash Gayah on Dec 19, 2016
Content may be subject to copyright.
AN EXPLORATORY ANALYSIS OF SIGNAL COORDINATION IMPACTS ON THE
MACROSCOPIC FUNDAMENTAL DIAGRAM
By
Jan-Torben Girault
ETH Zurich
Institute for Transport Planning and Systems
Ch. en Verney 12 cp 32, 1088 Ropraz, Switzerland
phone: +41 79 273 75 61
giraultj@student.ethz.ch
Vikash V. Gayah*
Department of Civil and Environmental Engineering
The Pennsylvania State University
231L Sackett Building
University Park, PA 16802
phone: 814-865-4014
gayah@engr.psu.edu
S. Ilgin Guler
Department of Civil and Environmental Engineering
The Pennsylvania State University
403 Sackett Building
University Park, PA 16802
phone: 814-867-6210
iguler@engr.psu.edu
Monica Menendez
Director of Research Group “Traffic Engineering”
Institute for Transport Planning and Systems
ETH Zurich
HIL F 37.2 Stefano-Franscini-Platz 5
8093, Zurich, Switzerland
phone: +41 44 633 66 95
monica.menendez@ivt.baug.ethz.ch
*Corresponding Author
November 2015
Word count: 7,813 (6,063 + 1,750 figures and tables)
Girault et al. 1
ABSTRACT
The Macroscopic Fundamental Diagram (MFD) of urban traffic is a recently developed tool to
model and study large-scale urban traffic networks. The MFD of a network is significantly
impacted by various properties of the network, including: signal green times, offsets, block
lengths, travel speeds, and routing behaviors. However, the understanding of the impact of
offsets (otherwise known as signal coordination) is limited to one-dimensional networks. The
impacts of coordination on the MFDs of two-dimensional networks are not well understood yet.
To shed some light on this topic, this study examines the impacts of signal coordination on an
idealized grid network using both an analytical method, and a microscopic traffic simulator
(AIMSUN). Seven coordination strategies are considered, including: simultaneous, alternating,
double alternating, one-dimensional green wave, two-dimensional green wave, MAXBAND, and
random. In general, the impacts of coordination are highly sensitive to the cycle length of the
network. The results also reveal that poor coordination can significantly decrease network
capacity and free flow travel speed. However, good coordination offers little advantage over
simultaneous offsets, even with directional demands traveling in the prioritized directions. The
reason is that benefits provided to vehicles in the prioritized direction are offset by disadvantages
to vehicles traveling in the non-prioritized directions. These results suggest that the insights from
other idealized simulations with simultaneous offsets are generalizable to more realistic
situations with better coordination strategies.
Girault et al. 2
INTRODUCTION
The introduction of the Network or Macroscopic Fundamental Diagram (MFD) of urban traffic
has provided researchers with a new tool to measure aggregate traffic patterns across dense,
spatially compact networks. The MFD provides a relationship between average vehicle flow and
average traffic density when measured across the entire network. While such relationships are
not new in the research literature (1-3), only recently has this tool been proven to exist in practice
through field tests using empirical data (4-6). These field tests have led to renewed interest in
macroscopic network traffic modeling due to the implications for the development of real-time
network-wide traffic control strategies. Such strategies include congestion pricing (7-15),
perimeter gating (10-16), network-wide routing (17; 18), and parking controls (19; 20).
A general lack of available empirical data on urban streets means that such field tests are
relatively rare. Thus, the majority of work on the MFD relies on simulation studies or theoretical
analyses of abstract networks. These simulation and theoretical studies have been used to assess
various aspects of MFDs. For example, simulation studies of idealized networks have revealed
inherent instabilities in networks that cause unpredictable and inefficient behavior when
congested (21-24), and that adaptive driving (25-27) and adaptive signal control (28; 29) can
help alleviate these negative effects.
Recent studies have used the MFD as a tool to measure the impacts of various operational
strategies or infrastructural changes on traffic through simulations of idealized networks. Knoop
et al. considered how road layout influences the shape of the MFD, and thus the performance of
the network (30). Muhlich et al. considered how the presence of hierarchical arterial streets on a
network impacted congestion patterns and the resulting MFD patterns in a network (31).
Ortigosa et al. used the MFD and a related relationship between trip completion rate and
accumulation—the Network Exit Function (NEF)—to compare street networks made up of one-
way and two-way streets (32). However, most existing simulation efforts, especially those that
study idealized network structures, assume that adjacent traffic signals are not coordinated within
the simulation environment. This assumption is typically made because it is the simplest choice
and it helps remove a degree of freedom from the simulation efforts. These results could be
questionable if the MFDs are significantly impacted by signal coordination.
Indeed, analytical studies using variational theory (VT) suggest that signal coordination
significantly influences the shape and form of an MFD (33-35). The VT method represents the
entire network by a single, infinitely long one-dimensional corridor and considers the flow-
density relationship of vehicles traveling in one direction along that corridor. For this reason,
good coordination has been assumed to always have a large positive effect on network operations
(i.e., the MFD) since it is relatively easy to provide coordination for a single travel direction.
However, this method ignores the effects that coordination may have on the opposite travel
direction. This is important as the benefits provided to vehicles traveling in one direction may be
canceled out by disbenefits to vehicles traveling in the opposite direction. Furthermore, networks
typically have two dimensions and this is not accounted for in the single-corridor representation.
As we will show, failure to account for this might misrepresent the impacts of coordination
provided in just one dimension.
The purpose of this study, therefore, is to examine the impacts of traffic signal
coordination on the MFD of a two-dimensional network. The VT method of cuts (33) is used to
obtain some analytical insights on this issue. Then, a battery of simulation scenarios are
Girault et al. 3
performed using networks of various sizes, seven signal coordination strategies, and multiple
demand patterns to confirm these findings. The results reveal that poor coordination can
significantly reduce free flow speeds and maximum capacities, but good coordination may have
little to no impact at the network-wide level when compared to common strategies like
simultaneous signal timings.
The rest of this paper is organized as follows. We first describe the analytical and the
simulation frameworks, coordination strategies, and demand patterns considered. Then, we
describe the methodology and metrics used to assess the impacts on the MFD. Next, we describe
the analytical and simulation results, as well as the insights obtained. Finally, we provide some
discussion and concluding remarks.
ANALYTICAL FRAMEWORK
We apply the VT method of cuts (33) to analytically derive the MFD of a two-dimensional
network while considering the impacts to vehicles traveling in both directions. Without loss of
generality, we abstract a bi-directional grid network into four unique corridors: south to north
(STN), north to south (NTS), west to east (WTE), and east to west (ETW). The original VT
method (derived for a single corridor) is applied to each of these corridors to obtain each
corridor’s MFD, and the MFD of the network is then approximated by the average of these four
corridor-level MFDs.
The corridors are not independent and this dependence must be accounted for before
developing the corridor-level MFDs. Corridors representing the same travel dimension (e.g.,
ETW and WTE) will have signals with similar cycle lengths and green times, and the offsets of
these corridors will also be related. The offset in one direction is equivalent to the difference
between the cycle length and the offset in the opposing direction. Corridors in perpendicular
dimensions (e.g., ETW and NTS) are also related, either due to the coordination strategy
assumed or the fact that different strategies can be used in different dimensions.
FIGURE 1 provides two examples of this method applied to a fictitious square grid
network with 200 m links, 90 second cycle length and 40 seconds of effective green time
provided to each direction. In all cases, a triangular fundamental diagram was assumed with free-
flow speed 50 km/hr, capacity 1800 veh/hr and jam density 125 veh/km. Each example
represents a different signal coordination strategy. FIGURE 1a shows that the corridor-level
MFDs, and thus the network-wide MFD, are the same when signals have no offset. We refer to
this as the simultaneous coordination case since all signal timings change simultaneously,
although it could also represent no coordination. FIGURE 1b shows the case where a green-wave
is provided to the STN direction. Notice that this green-wave gives the STN direction a larger
free-flow speed (slope of the increasing branch of the MFD) and causes a much lower free-flow
speed for the NTS direction. The MFD of the WTE and ETW directions remain unchanged,
since they are assumed to maintain simultaneous offsets. Also notice that even though the
capacities of the corridor-level MFDs do not change (i.e., remain 800 veh/hr), the overall
network MFD exhibits a decrease in capacity of about 50 veh/hr (6%). This occurs because the
capacities on the corridors are observed for different ranges of density (i.e., the critical density on
one corridor might be observed when another corridor is congested or free-flowing, and this
lowers the average vehicle flow across the whole network for that density). The average free-
Girault et al. 4
flow speed decreases from 22.22 km/hr in the simultaneous coordination case to just 20.9 km/hr
in the STN green-wave case (decrease of 6%).
a) b)
FIGURE 1. Analytically derived MFDs for: a) simultaneous coordination; and, b) green-wave in
the STN direction.
This analytical method was used to study the impacts of signal coordination on two-
dimensional networks; the results will be discussed in a later section. Unfortunately, the
analytical method relies on several assumptions that might not be realistic. For one, it ignores the
impacts of vehicle acceleration and deceleration along the corridor, which can significantly
impact travel speeds and the MFD. The analytical method also assumes that the vehicle density
is the same across all four corridors at any time. However, poor progression in one direction can
result in a larger density of vehicles in that direction compared to another with better
coordination. To address these issues, simulations are also carried out.
SIMULATION FRAMEWORK
To examine how these more realistic impacts might change the general trends and insights
observed from the analytical model, micro-simulation tests were performed. The remainder of
this section describes the simulated network, the signal coordination strategies, and the demand
patterns considered.
Network
A symmetric, 10x10 square grid network was considered. This was chosen because grid
structures are highly conducive to signal coordination. Thus, differences in the MFD caused by
signal coordination should be readily apparently on this type of networks. It also mimics the
“two-corridor” approach of the analytical framework. Two-way streets with two travel lanes per
direction were used to create all blocks. Each of the links were 200 meters long arterial roads
with capacity of 1800 vehicles per hour and speed limit of 50 kilometers per hour. These
parameters yielded a free-flow travel time of 14.4 seconds between two consecutive
intersections. The total network length was 176 lane-km.
Girault et al. 5
Origin and destinations were coded at the midblock locations of all links as opposed to
the intersection locations. This was done to facilitate a more realistic representation of vehicles
entering and exiting the network from driveways within the network, and to minimize unrealistic
vehicle maneuvers at intersections. Origins and destinations were also created at
upstream/downstream ends of the peripheral links to consider external traffic. A total of 220
origins/destination were created.
Demand
The traffic demand in a network has a significant influence in the performance of the network,
especially when traffic signal coordination is considered. In this research, the network is
simulated starting from an empty state until a complete gridlock is reached. Various demand
patterns are implemented to see if the combination of directional demand and signal coordination
would exacerbate the impacts of signal coordination on the MFD. Four different demand patterns
were studied:
• Uniform demand
• One-dimensional unidirectional demand
• One-dimensional bidirectional demand
• Two-dimensional unidirectional demand
In the uniform demand strategy, the same number of trips is assigned to all origin-
destination pairs. This uniform demand pattern is common in simulation studies of the MFD,
since MFDs are more likely to exist when demand is distributed evenly in a network. In the one-
dimensional unidirectional demand, external origins on one of the sides of the network (e.g., the
south side) generate 20 times more trips than all other origins. This represents a directional
demand case in which more vehicles enter the network from one side than the others; see
FIGURE 2a. In the two-dimensional unidirectional demand, origins on two adjacent edges of the
network generate 10 times more demand than all other origins; see FIGURE 2a. This could
represent a case where the network is located between two large demand generators. Lastly, in
the one-dimensional bidirectional demand, external origins on two opposing directions generate
10 times more trips than others; see FIGURE 2a. In all cases, the same network loading strategy
was applied. The network was initially empty and the demand increased at steady 10-minute
intervals until the network completely gridlocked to ensure that the entire MFD was observed.
Girault et al. 6
(a) Directional demand patterns
(b) Directional coordination strategies
(c) Analysis metrics
FIGURE 2. Illustration of: a) directional demand patterns; b) coordination strategies; and, c)
analysis metrics.
Girault et al. 7
Signal timing and coordination
Traffic signals were implemented at the intersection of all links. Since the focus of this paper is
on signal coordination, here we only consider fixed traffic signal timings and do not consider
actuated or adaptive traffic signal control. A simple two-phase signal-timing scheme was
implemented with one phase serving north-south traffic and the other phase serving east-west
traffic. During each phase, all movements (left, through and right) were allowed. Thus, left turns
were served in a permitted fashion (i.e., left-turning vehicles had to yield to through-moving
vehicles at intersections). Right-turn-on-red maneuvers were forbidden. A total of 100
intersections were included. Three cycle lengths were considered: 60 seconds, 90 seconds, and
120 seconds. In each case, a constant yellow time of 3 seconds and all-red time of 2 seconds was
implemented during each phase change. The remaining time was divided evenly to provide green
time to each of the two competing directions.
This study considered common signal coordination strategies that were noted in the research
literature (36). In all, seven different signal coordination strategies were tested as a part of this
work:
• Simultaneous
• Alternating
• Double alternating
• One-dimensional green wave
• Two-dimensional green wave
• MAXBAND
• Random
These strategies are implemented by adjusting the offsets between the time the green
signal starts at adjacent intersections. In the simultaneous strategy, all offsets are set to 0 (i.e.,
signals are not coordinated). Thus, all east-west (or north-south) green phases occur at the same
time for all intersections within the network. This strategy is the simplest strategy and likely to
occur in networks where coordination is not considered, plus is typically used in the literature for
the analysis of MFDs under multiple conditions (31; 32). This is also used as the baseline
scenario in this paper. In the alternating strategy, adjacent signals have offsets that are equal to
one-half of the cycle length. Thus, on any given row or column of the grid, every other signal
will start its east-west phase at the same time, while the remainder start the north-south phase.
The double alternating strategy is similar to the alternating strategy; except intersections are
coupled in adjacent pairs and then the alternating strategy is applied to the pairs. These two
strategies are commonly used methods to provide two-directional coordination on arterial streets
(37). The one-dimensional green wave strategy uses optimal offsets equal to the free-flow travel
time (about 14 seconds) to provide a green wave in a single dimension (e.g., south to north); see
FIGURE 2b. The two-dimensional green wave is the same as the one-dimensional version except
that the green wave is provided in the two perpendicular dimensions simultaneously (e.g., west
to east and south to north); see FIGURE 2b. The MAXBAND strategy is an optimization
algorithm that selects offsets to provide the largest bandwidth to vehicles traveling in opposite
directions along the same dimension (e.g., south to north and north to south) (38; 39); see
FIGURE 2b. The offsets are determined using a mixed integer linear program that tries to
maximize the bandwidth of the green wave in both directions while accounting for the
limitations of the available signal timings. Lastly, the random strategy uses offsets generated
Girault et al. 8
from a uniform distribution between 0 and the cycle length. This strategy might replicate a
network that pays absolutely no attention to coordination or signal interactions. Only the
simultaneous, alternating, 1D and 2D green-waves were considered in the analytical framework
as the formulas provided in (33) are only applicable when regular offsets are maintained between
adjacent signals.
Note also that the STN direction is chosen as the high demand direction in the one-
dimensional unidirectional demand pattern. In the one-dimensional bidirectional demand pattern,
the STN and NTS directions have higher demands. Lastly, in the two-dimensional unidirectional
demand pattern, the STN and WTE directions have higher demands. Signal coordination
strategies that favor a particular direction(s) of movement (e.g., the green wave and MAXBAND
strategies) prioritize these directions. In other words, the signal coordination and demand
patterns are in sync.
Analytical and Simulation Metrics
Flow and density data were aggregated at 5-minute intervals for each of the simulation iterations.
These flow and density values were obtained by applying the generalized definitions of Edie (40)
to individual vehicle trajectories outputted from the simulation software. For each scenario, the
data from the 10 simulation iterations were aggregated together to obtain the MFD representing
overall network behavior. These MFDs were directly compared to assess the impacts of signal
coordination. Three specific features of the MFDs were considered in these comparisons. The
first was the capacity of the network, measured as the largest flow value observed on the MFD.
The capacity provides an indication of the maximum productivity of the network. We would
naively expect capacity to increase with better coordination as vehicles need to stop fewer times.
The second metric was the free flow speed, measured as the slope of the left-hand (or free flow)
side of the MFD at capacity. Intuitively, the free flow speed would be the most influenced by
signal coordination as good coordination is associated with fewer stops, and thus a larger
proportion of time traveling at free flow speed. The third metric was the backward wave speed,
i.e., the slope of the right-hand (or congested) branch of the MFD. All of these metrics are
illustrated in FIGURE 2c. Each metric was computed for each of the 10 simulation iterations for
each scenario. The mean and the variation of these metrics were used to facilitate statistical tests
of equality. For the analytical tests, these metrics were obtained from the MFD directly.
Note that congested states are characterized by instabilities, which result in scattered and
chaotic MFDs, especially in simulations (22; 23). Thus, although we look at the congested
branch as stated above, we focus more on the free-flow and capacity states since these are more
reproducible and desirable.
RESULTS
This section describes the results of both the analytical and the simulation studies to reveal the
impacts of signal coordination on the MFD.
First, a visual inspection of the MFDs generated from the simulation is conducted. FIGURE 3
provides the simulated MFDs for uniform demand patterns for the considered cycle lengths and
coordination strategies. Two forms are provided: flow-density and speed-density. The latter more
Girault et al. 9
readily reveals changes to free-flow speed observed in the simulations. A cursory visual analysis
of these MFDs reveals that the MFDs are rather consistent across the different signal
coordination strategies. No significant differences are visually apparent in the free-flow or
capacity portions of the MFDs presented. Some scatter is observed for the congested branch, but
the changes in absolute values of the backward wave speed are minimal. This trend might make
sense for the uniform demand pattern, since coordination is typically provided to favor
significant vehicle movement in one (or more) direction(s). Therefore, it might not be surprising
that signal coordination does not have significant impacts on the MFD when demand patterns are
completely uniform.
a)
0
50
100
150
200
250
300
350
400
450
500
0 20 40 60 80 100 120 140 160
FLOW (VEH/HR)
DENSITY (VEH/KM)
Cycle = 60 sec
0
50
100
150
200
250
300
350
400
450
500
0 20 40 60 80 100 120 140 160
FLOW (VEH/HOUR)
DENSITY (VEH/KM)
Cycle = 90 sec
0
50
100
150
200
250
300
350
400
450
500
0 20 40 60 80 100 120 140 160
FLOW (VEH/HOUR)
DENSITY (VEH/KM)
Cycle = 120 sec
0
50
100
150
200
250
300
350
400
450
500
0 20 40 60 80 100 120 140 160
Flow (Veh/hour)
Density (veh/km)
Cycle = 60 sec Simultaneous
Alternating
Double Alternating
Green Wave
Green Wave 2D
Maxband
Random
Girault et al. 10
b)
FIGURE 3 Simulated MFDs for uniform demand pattern in: a) flow-density format; and b) speed-
density format.
Surprisingly, the MFDs for other demand patterns tend to follow the same basic shape
even when there is directional demand, and coordination is provided to favour that direction.
There are two mechanisms that exist to explain this behaviour. The first is that benefits provided
to vehicles traveling in one direction are associated with disbenefits (more stops and lower travel
speeds) to vehicles traveling in the opposite direction. Visual inspections of the simulations
reveal more stopping in non-prioritized directions. The second is that since prioritized, high-
demand directions serve more vehicles, they become congested more quickly than the non-
prioritized, low-demand directions. Since the different directions become congested at different
times, the capacity of the network shown in the MFD reflects different states across the different
directions: free-flow states in the low demand directions and congested states in the high demand
directions. Thus, any additional flows in the prioritized directions are associated with lower
flows in the non-prioritized directions, which yields total network capacities that are not much
different.
To better understand the impacts of different signal coordination strategies, TABLE 1,
TABLE 2, and TABLE 3 provide the metrics of interest (capacity flow, free flow speed, and
backward wave speed) for all network scenarios considered in this study. The top three rows in
each table provide the corresponding metric for the simultaneous coordination strategy, while the
remaining rows show the difference between each strategy and the simultaneous value. The
value in parentheses indicates the relative difference. For example, in TABLE 1, the 10 (2.4%)
entry means that for a 60 second cycle length and uniform demand pattern, the simulated
alternating coordination strategy yielded a maximum capacity of 434 veh/hr (equal to 10+424)
0
5
10
15
20
25
30
35
40
0 5 10 15 20
SPEED (KPH)
DENSITY (VEH/KM)
Cycle = 60 sec
0
5
10
15
20
25
30
35
40
0 5 10 15 20
SPEED (KPH)
DENSITY (VEH/KM)
Cycle = 90 sec
0
5
10
15
20
25
30
35
40
0 5 10 15 20
SPEED (KPH)
DENSITY (VEH/KM)
Cycle = 120 sec
0
50
100
150
200
250
300
350
400
450
500
0 20 40 60 80 100 120 140 160
Flow (Veh/hour)
Density (veh/km)
Cycle = 60 sec Simultaneous
Alternating
Double Alternating
Green Wave
Green Wave 2D
Maxband
Random
Girault et al. 11
and this is a 2.4% change from the simulated simultaneous strategy. Bolded entries for the
simulated scenarios represent values that are significantly different from the simultaneous
coordination strategy using a statistical t-test to compare the means at a 95% significance level.
For the capacity values (TABLE 1), the first thing we notice is that the analytical results
are overly optimistic compared to the simulated results. For example, for the simultaneous
strategy, the analytical capacities are 1.8 to 1.9 times higher than those observed in the
simulations with a uniform demand. This is not surprising, as the analytical method does not
account for the variability of driver behavior (e.g., acceleration and deceleration maneuvers) and
density differences in the network. That being said, both in the analytical and the simulation
results, we see that the capacity tends to increase with the cycle length for the simultaneous
strategy. This is reasonable, as the proportion of time lost for vehicle movement decreases with
the cycle length. Thus, longer cycles are expected to have higher capacity if no queue spillbacks
arise. Note that both the analytical model and simulation framework account for the impacts of
these queue spillbacks in their results. When considering the rest of the coordination strategies,
we can see that many entries are significantly different from those of the simultaneous strategy.
The random coordination almost always provides a statistically lower capacity, as expected,
since random offsets are a worst-case scenario in which coordination is completely ignored.
TABLE 1 also reveals that impacts of the coordination strategies on network capacity are highly
sensitive to the cycle length. For example, for all simulated demand scenarios the alternating and
double alternating strategies provide modest (up to 5%) capacity improvements for 60-second
cycle lengths as opposed to 90 or 120 second cycle lengths, which generally provide a capacity
decrease. This is not unexpected, as coordination is a complicated process that requires signal
timings to be in sync with travel times along individual links in the network. FIGURE 4 provides
an illustrative example, which shows that the alternating strategy is beneficial for 60-second
cycle lengths, but not 120 second cycle length. The former scenario requires vehicles in both
directions to wait just small amounts at signal red periods while the latter requires much longer
waits. The additional queues that are created as a result of these longer waits can reduce
throughput at the intersections.
Unfortunately, the analytical method does not yield any changes in capacity for the
alternating strategy. This is because the analytical capacity for this case is bounded by the green
time available for movement; i.e., the free-flow and congested “cuts” do not influence capacity
in this case. Additional tests suggests lower capacity would be observed in the analytical method
if the link lengths were decreased, so this is likely due to the set of parameters chosen. For the
1D and 2D green wave, the analytical method consistently predicts a reduction in network
capacity as the cycle length increases (with drops up to 11% for the 120 second cycle length).
This is slightly different from the results obtained from simulation that not only follow a
different pattern for different cycle lengths, but show a maximum drop of less than 4%. As a
matter of fact, none of the directional priority strategies (1D and 2D green wave, MAXBAND)
generally offers significantly different network capacities in the simulations, even when demands
are directional. Even the changes that are statistically significant tend to be either negative or
very small. This is not surprising: although the directional strategies provide priority to one (or
more) directions, this is simultaneously associated with very poor progression to the
disadvantaged directions. It appears the higher flows in one directions are compensated by lower
flows in the other directions for a net neutral change in overall network capacity; this trend is
confirmed by the analytical model. Even when the coordination strategy tries to account for both
Girault et al. 12
directions (as MAXBAND does), still these results hold,implying that it is very difficult to
provide effective coordination in both directions.
TABLE 1. Capacity metrics
Cycle
(sec)
Analytical
Uniform
1D unidirectional
1D bidirectional
2D unidirectional
Capacity (veh/km)
Simultaneous
60
750
424.4
429.9
438.7
429.9
90
800
427.4
439.1
446.2
444.9
120
825
434
449.2
452.7
456.3
Change in capacity (veh/km), and percentage difference from simultaneous in
parenthesis (bold values indicate statistically significant changes)
Alternating
60
0 (0%)
10 (2.4%)
21 (4.9%)
18 (4.1%)
23 (5.4%)
90
0 (0%)
-33 (-7.8%)
-45 (-10.5%)
-52 (-11.9%)
-42 (-9.8%)
120
0 (0%)
-86
(-19.8%)
-120 (-26.7%)
-113 (-25%)
-111 (-24.3%)
Double
Alternating
60
N/A
8 (1.9%)
24 (5.6%)
15 (3.4%)
20 (4.7%)
90
N/A
-4 (-0.9%)
0 (0%)
-6 (-1.4%)
1 (0.2%)
120
N/A
-33 (-7.6%)
-25 (-5.6%)
-35 (-7.7%)
-42 (-9.2%)
Green Wave
60
0 (0%)
-5 (-1.2%)
4 (0.9%)
-5 (-1.1%)
-1 (-0.2%)
90
-13 (-1.6%)
-2 (-0.5%)
10 (2.3%)
-5 (-1.1%)
1 (0.2%)
120
-79 (-9.6%)
-8 (-1.8%)
-3 (-0.7%)
-4 (-0.9%)
-11 (-2.4%)
Green Wave
2D
60
0 (0%)
-7 (-1.6%)
0 (0%)
-8 (-1.8%)
8 (1.9%)
90
-2 (-0.3%)
-4 (-0.9%)
1 (0.2%)
-8 (-1.8%)
1 (0.2%)
120
-90 (-10.9%)
-16 (-3.7%)
-10 (-2.2%)
-15 (-3.3%)
-18 (-3.9%)
MAXBAND
60
N/A
2 (0.5%)
11 (2.6%)
6 (1.4%)
15 (3.5%)
90
N/A
-6 (-1.4%)
0 (0%)
-6 (-1.4%)
-3 (-0.7%)
120
N/A
-18 (-4.1%)
-15 (-3.3%)
-20 (-4.4%)
-28 (-6.1%)
Random
60
N/A
-9 (-2.1%)
-23 (-5.4%)
0 (0%)
-16 (-3.7%)
90
N/A
-16 (-3.8%)
-25 (-5.8%)
-21 (-4.8%)
-16 (-3.7%)
120
N/A
-28 (-6.5%)
-35 (-7.8%)
-36 (-8%)
-55 (-12.1%)
Girault et al. 13
(a) 60 second cycle length
(b) 120 second cycle length
FIGURE 4. Illustrative time-space diagram showing the benefits of alternating coordination for (a)
60 second cycle lengths as compared to (b) 120 second cycle lengths.
Changes in free flow travel speed (TABLE 2) show the same basic trends and patterns,
although the percent changes of the free flow speeds are generally much larger than the changes
in capacity. This is likely due to the low free flow speeds observed (about 20 kph). Notice that in
this case, the analytical method provides much more consistent results compared to those
obtained from the simulations for the simultaneous coordination strategy, although they follow
the opposite pattern with respect to cycle lengths. Most of the significant changes in network free
flow speeds occur for the alternating, double alternating, and random strategies. Moreover, the
analytical method yields higher speed changes than the simulations in most cases. Again, most of
the strategies appear to provide lower free flow speeds across the entire network. In particular,
the random offset strategy provides the lowest speeds, as expected. The only strategies that
appear to consistently increase the network speed are the alternating and double alternating
strategies for a cycle length of 60 seconds. This is not surprising as they are both typical
strategies employed to improve coordination. However, these results are not general and the
improvement shown here is due to the specific combinations of the parameters used in these
tests. The results from a 16 x 16 network (not shown here) reveal that these improvements are
Girault et al. 14
not always observed. The directional coordination strategies again do not offer significant
benefits when observed network-wide as compared to the other strategies, even when demands
are directional as well. In general, coordination appears to have little effect on the network speed,
even for different, unbalanced demand scenarios. This again, is most likely due to the fact that
while coordination increases the travel speed along one direction, it increases the number of
stops along the opposite direction and the overall network speed remains unchanged.
TABLE 2. Free-flow speed metrics
Cycle
(sec)
Analytical
Uniform
1D
unidirectional
1D
bidirectional
2D
unidirectional
Speed (kph)
Simultaneous
60
22.6
22.6
21.1
21.5
20.9
90
23.5
20.2
20.9
19.8
20.6
120
23.9
18.9
18.8
18.5
18
Change in speed (kph), and percentage difference from simultaneous in parenthesis
(bold values indicate statistically significant changes)
Alternating
60
1.4 (6.2%)
0.5 (2.2%)
1.8 (8.5%)
1.9 (8.8%)
2.4 (11.5%)
90
-7.5 (-31.8%)
-2.9 (-14.4%)
-3.6 (-17.2%)
-3.9 (-19.7%)
-4.8 (-23.3%)
120
-11.9 (-49.8%)
-4.5 (-23.8%)
-4.8 (-25.5%)
-4.6 (-24.9%)
-5.1 (-28.3%)
Double
Alternating
60
N/A
1.6 (7.1%)
3.1 (14.7%)
2.8 (13%)
3.3 (15.8%)
90
N/A
-0.2 (-1%)
-0.9 (-4.3%)
0.3 (1.5%)
-1.5 (-7.3%)
120
N/A
-2.3 (-12.2%)
-2.5 (-13.3%)
-2.3 (-12.4%)
-2.2 (-12.2%)
Green Wave
60
-1.1 (-5%)
-1 (-4.4%)
0.6 (2.8%)
0.2 (0.9%)
-0.2 (-1%)
90
-0.5 (-2%)
-0.5 (-2.5%)
-0.7 (-3.3%)
1 (5.1%)
-1.4 (-6.8%)
120
-1.3 (-5.2%)
0.3 (1.6%)
0.2 (1.1%)
0.9 (4.9%)
0.6 (3.3%)
Green Wave
2D
60
-2.3 (-10.2%)
-1.2 (-5.3%)
-1 (-4.7%)
-1.4 (-6.5%)
0.3 (1.4%)
90
-0.8 (-3.5%)
-0.5 (-2.5%)
-0.7 (-3.3%)
0.2 (1%)
0.9 (4.4%)
120
-2.5 (-10.4%)
-0.9 (-4.8%)
-0.3 (-1.6%)
-1.4 (-7.6%)
-0.3 (-1.7%)
MAXBAND
60
N/A
0.5 (2.2%)
0.8 (3.8%)
0.6 (2.8%)
1.3 (6.2%)
90
N/A
-1.2 (-5.9%)
-2.1 (-10%)
-1.2 (-6.1%)
-1.5 (-7.3%)
120
N/A
-1.8 (-9.5%)
-1.1 (-5.9%)
-1.1 (-5.9%)
-0.8 (-4.4%)
Random
60
N/A
-0.9 (-4%)
-4.9 (-23.2%)
-0.4 (-1.9%)
-3.1 (-14.8%)
90
N/A
-1.3 (-6.4%)
-3.4 (-16.3%)
-2.2 (-11.1%)
-2.5 (-12.1%)
120
N/A
-2.7 (-14.3%)
-2.9 (-15.4%)
-2.9 (-15.7%)
-2.8 (-15.6%)
Changes in the speed of the congested branch of the MFD (TABLE 3) complement the
capacity and free-flow speed observations explained above. As FIGURE 3 shows, jam density
values remain very similar across coordination strategies (the same is true for the other demand
patterns, not shown here due to space constraints). Hence, one would expect that the backward
wave speed should change if the free-flow speed changes and the capacity is held constant.
TABLE 3 shows that these changes can be rather drastic in terms of percentage, but this is due to
the fact that the absolute values for the backward wave speed are rather small (less than -5.2kph
for all simulations with the simultaneous coordination). In this case, the analytical method yields
Girault et al. 15
a faster speed for the backward wave in the simultaneous coordination strategy, and bigger
changes for the other strategies, compared to the results observed in the simulations. However,
these results are driven by the much higher network capacities predicted with the analytical
approach.
TABLE 3. Slope of congested branch metrics
Cycle
(sec)
Analytical
Uniform
1D
unidirectional
1D
bidirectional
2D
unidirectional
Slope of congested branch (kph)
Simultaneous
60
-11.9
-3.4
-3.4
-3.5
-3.4
90
-7.8
-4.5
-4.6
-4.3
-4.3
120
-6.0
-5.2
-4.6
-5.1
-5
Change in speed of congested branch (kph), and percentage difference from
simultaneous in parenthesis (bold values indicate statistically significant changes)
Alternating
60
-2.4 (-20.3%)
-1.2 (35.3%)
-1 (29.4%)
-1 (28.6%)
-1 (29.4%)
90
7.5 (96.5%)
0.1 (-2.9%)
0.8 (-23.5%)
0.2 (-5.7%)
0.2 (-5.9%)
120
5.9 (98.7%)
0.6 (-17.6%)
0.7 (-20.6%)
0.8 (-22.9%)
0.5 (-14.7%)
Double Alternating
60
N/A
-0.7 (20.6%)
-0.9 (26.5%)
-0.7 (20%)
-0.8 (23.5%)
90
N/A
-0.8 (23.5%)
-0.6 (17.6%)
-0.5 (14.3%)
-0.9 (26.5%)
120
N/A
-0.2 (5.9%)
-0.1 (2.9%)
0 (0%)
0 (0%)
Green Wave
60
-4.8 (-40.8%)
-0.3 (8.8%)
-0.2 (5.9%)
-0.1 (2.9%)
-0.1 (2.9%)
90
-1.2 (-15.1%)
-0.8 (23.5%)
-1 (29.4%)
-0.8 (22.9%)
-1.2 (35.3%)
120
-2.2 (-37.2%)
-1.3 (38.2%)
-1.2 (35.3%)
-1.4 (40%)
-1 (29.4%)
Green Wave 2D
60
-5.2 (-43.5%)
-0.5 (14.7%)
-0.3 (8.8%)
-0.4 (11.4%)
-0.4 (11.8%)
90
-0.7 (-9.4%)
-0.9 (26.5%)
-0.9 (26.5%)
-0.7 (20%)
-0.8 (23.5%)
120
0.8 (13.6%)
-1.3 (38.2%)
-1 (29.4%)
-0.7 (20%)
-1 (29.4%)
MAXBAND
60
N/A
-0.5 (14.7%)
-0.3 (8.8%)
-0.1 (2.9%)
-0.4 (11.8%)
90
N/A
-0.8 (23.5%)
-0.6 (17.6%)
-0.5 (14.3%)
-0.6 (17.6%)
120
N/A
-1.1 (32.4%)
-1.2 (35.3%)
-1.1 (31.4%)
-0.8 (23.5%)
Random
60
N/A
-0.2 (5.9%)
-0.3 (8.8%)
-0.2 (5.7%)
-0.2 (5.9%)
90
N/A
-0.4 (11.8%)
-0.3 (8.8%)
-0.2 (5.7%)
-0.4 (11.8%)
120
N/A
-0.4 (11.8%)
-0.1 (2.9%)
-0.1 (2.9%)
0 (0%)
CONCLUDING REMARKS
In this study, we examined the impacts of signal coordination on the macroscopic fundamental
diagram of a two-dimensional grid network. An analytical method was proposed to estimate the
impacts of signal coordination on two-dimensional networks abstracted as two unique
bidirectional corridors. This method is a significant improvement over existing analytical
methods, which abstract a network as a single unidirectional corridor. This method reveals the
negative impacts of coordination on other travel directions. Microsimulations tests in AIMSUN
supplemented the analytical model. Seven popular coordination strategies were considered:
Girault et al. 16
simultaneous, alternating, double alternating, one-dimensional green wave, two-dimensional
green wave, MAXBAND, and random. These were tested under various demand patterns:
completely uniform, one-dimensional unidirectional, one-dimensional bidirectional, and two-
dimensional unidirectional. The impacts of signal coordination on the MFD were measured by
the change in capacity, the change in free flow speed, and the change in backward wave speed.
Overall, our study confirmed that the impacts of coordination on the MFD are highly
dependent on the cycle length used for signal timings throughout the network. Strategies that
offer modest increases in an analysis metric for a given cycle length might have the opposite
impact for another. Thus, recommendations for the optimal coordination strategy cannot be made
for general networks, since this choice will depend on a variety of network specific factors,
including: link length, speed limit, signal phasing scheme, and cycle length. However, a few
generic trends were observed. For one, random offsets always result in reduced performance as
measured by the MFDs. This is an important finding as some networks either do not account for
coordination at all (which could lead to reduced network performance) or have offsets based on
other factors that might appear random from a vehicle’s perspective. Such situations should be
avoided, or at least coordinated with vehicle movement to not hinder traffic performance.
Another general trend appears to be that directional priority strategies (such as the green
wave and MAXBAND strategies) do not provide superior network performance, even when
demand is directional. This occurs because the travel advantages obtained by vehicles traveling
in the preferred directions appear to be mitigated by the poor signal timing encountered by
vehicles traveling in the opposite and perpendicular directions, which are not prioritized. For the
most part, these strategies perform about the same as the simultaneous offset strategy, which was
used as the base case here, at least for the battery of simulation tests performed. This is also an
important finding, as much of the existing research on MFDs uses simultaneous coordination for
simplification purposes. It is now clear that such an assumption is not limiting, and using more
complex coordination strategies would not necessarily lead to much different results.
The analytical method was shown to consistently over-predict network capacities, leading
to faster backward waves, compared to simulations using a uniform demand. However, it yielded
relatively similar results in terms of free-flow speeds, which might be useful for many
applications. In addition, as with the simulations, it did not show any consistent or generalizable
pattern for the different coordination strategies. Instead, the percent changes in capacity were
still modest, and free-flow speeds tended to decrease with any coordination.
Our study only focused on regular square grid networks, since this is a “best case” for
signal coordination. Regular rectangular networks can be treated in a similar manner, and similar
results would be expected. The presence of irregular or disruptive elements will serve to make
coordination more difficult and thus reduce any of the (small) benefits that may be achieved
through signal coordination.
Of course, this study is not without its limitations. Only a discrete set of scenarios were
considered in the experimental design. Since coordination is highly sensitive to cycle lengths and
other factors, it is possible that other interesting insights would be observed for different sets of
parameters. Still, the general insights discussed here should remain. This study also focused on
fixed signal timings; however, actuated to adaptive traffic signals serve to improve flows and
vehicle progression in networks (28; 29). Thus, it is possible that the combination of
coordination with these strategies might generate new findings. We also only considered
Girault et al. 17
networks with increasing demands that were simulated to gridlock. Tests with non-monotonic
demands that decrease during the end of a rush might reveal which coordination strategies are
best able to serve the transitions between congested and free flow states. The MFD can be used
as a tool to reveal these changes since these demand patterns are typically characterized by
hysteresis loops in the MFD. Future work will consider the impact of coordination on these
hysteresis loops to see which coordination strategies are most robust to increasing and decreasing
demand patterns. To improve the analytical approach, it might be worthwhile to investigate
other aggregation methods across the MFDs from the individual corridors, as well as alternative
ways to account for the dependence between corridors.
ACKNOWLEDGMENTS
The authors would like to thank Javier Ortigosa for his thoughtful discussions and contributions
to this work.
REFERENCES
[1] Godfrey, J. The mechanism of a road network. Traffic Engineering & Control, Vol. 11, No.
7, 1969, pp. 323-327.
[2] Mahmassani, H., J. Williams, and R. Herman. Performance of urban traffic networks.In
Transportation and Traffic Theory (Proceedings of the Tenth International on Transportation
and Traffic Theory Symposium, Cambridge, Massachusetts), NH Gartner, NHM Wilson, editors,
Elsevier, 1987.
[3] Mahmassani, H., J. C. Williams, and R. Herman. Investigation of network-level traffic flow
relationships: Some simulation results. Transportation Research Record: Journal of the
Transportation Research Board, Vol. 971, 1984, pp. 121-130.
[4] Geroliminis, N., and C. F. Daganzo. Existence of urban-scale macroscopic fundamental
diagrams: Some experimental findings. Transportation Research Part B: Methodological, Vol.
42, No. 9, 2008, pp. 759-770.
[5] Buisson, C., and C. Ladier. Exploring the impact of homogeneity of traffic measurements on
the existence of macroscopic fundamental diagrams. Transportation Research Record: Journal
of the Transportation Research Board, Vol. 2124, No. 1, 2009, pp. 127-136.
[6] Tsubota, T., A. Bhaskar, and E. Chung. Brisbane macroscopic fundamental diagram:
empirical findings on network partitioning and incident detection. Transportation Research
Record: Journal of the Transportation Research Board, Vol. in press, 2014.
[7] Geroliminis, N., and D. M. Levinson. Cordon pricing consistent with the physics of
overcrowding.In Transportation and Traffic Theory 2009: Golden Jubilee, Springer, 2009. pp.
219-240.
[8] Simoni, M., A. Pel, R. Waraich, and S. Hoogendoorn. Marginal cost congestion pricing based
on the network fundamental diagram. Transportation Research Part C: Emerging Technologies,
Vol. 56, 2015, pp. 221-238.
Girault et al. 18
[9] Zheng, N., R. A. Waraich, K. W. Axhausen, and N. Geroliminis. A dynamic cordon pricing
scheme combining the Macroscopic Fundamental Diagram and an agent-based traffic model.
Transportation Research Part A: Policy and Practice, Vol. 46, No. 8, 2012, pp. 1291-1303.
[10] Geroliminis, N. Increasing mobility in cities by controlling overcrowding.In, University of
California, Berkeley, 2007.
[11] Keyvan-Ekbatani, M., A. Kouvelas, I. Papamichail, and M. Papageorgiou. Exploiting the
fundamental diagram of urban networks for feedback-based gating. Transportation Research
Part B: Methodological, Vol. 46, No. 10, 2012, pp. 1393-1403.
[12] Keyvan-Ekbatani, M., M. Papageorgiou, and I. Papamichail. Urban congestion gating
control based on reduced operational network fundamental diagrams. Transportation Research
Part C: Emerging Technologies, Vol. 33, 2013, pp. 74-87.
[13] Haddad, J., and N. Geroliminis. On the stability of traffic perimeter control in two-region
urban cities. Transportation Research Part B: Methodological, Vol. 46, No. 9, 2012, pp. 1159-
1176.
[14] Haddad, J., M. Ramezani, and N. Geroliminis. Optimal perimeter control for two urban
regions with macroscopic fundamental diagrams: A model predictive control approach.
Intelligent Transportation Systems, IEEE Transactions on, Vol. 14, No. 1, 2012, pp. 348-359.
[15] Haddad, J., M. Ramezani, and N. Geroliminis. Cooperative traffic control of a mixed
network with two urban regions and a freeway. Transportation Research Part B:
Methodological, Vol. 54, 2013, pp. 17-36.
[16] Ortigosa, J., M. Menendez, and H. Tapia. Study on the number and location of measurement
points for an MFD perimeter control scheme: a case study of Zurich. EURO Journal on
Transportation and Logistics, Vol. 3, No. 3-4, 2014, pp. 245-266.
[17] Knoop, V., S. Hoogendoorn, and J. W. Van Lint. Routing strategies based on macroscopic
fundamental diagram. Transportation Research Record: Journal of the Transportation Research
Board, Vol. 2315, No. 1, 2012, pp. 1-10.
[18] Yildirimoglu, M., and N. Geroliminis. Approximating Dynamic Equilibrium Conditions
with Macroscopic Fundamental Diagrams.In Transportation Research Board 93rd Annual
Meeting, 2014.
[19] Cao, J., and M. Menendez. System dynamisc of urban traffic based on its parking-related-
states. Transportation Research Part B, in press, DOI 10.1016/j.trb.2015.07.018, 2015.
[20] Geroliminis, N. Cruising-for-parking in congested cities with an MFD representation.
Economics of Transportation, Vol. 3, 2015, pp. 156-165.
[21] Mazloumian, A., N. Geroliminis, and D. Helbing. The spatial variability of vehicle densities
as determinant of urban network capacity. Philosophical Transactions of the Royal Society A:
Mathematical, Physical and Engineering Sciences, Vol. 368, No. 1928, 2010, pp. 4627-4647.
[22] Gayah, V. V., and C. F. Daganzo. Effects of turning maneuvers and route choice on a
simple network. Transportation Research Record: Journal of the Transportation Research
Board, Vol. 2249, No. 1, 2011, pp. 15-19.
Girault et al. 19
[23] Daganzo, C. F., V. V. Gayah, and E. J. Gonzales. Macroscopic relations of urban traffic
variables: Bifurcations, multivaluedness and instability. Transportation Research Part B:
Methodological, Vol. 45, No. 1, 2011, pp. 278-288.
[24] Gayah, V. V., and C. F. Daganzo. Clockwise hysteresis loops in the macroscopic
fundamental diagram: an effect of network instability. Transportation Research Part B:
Methodological, Vol. 45, No. 4, 2011, pp. 643-655.
[25] Mahmassani, H. S., M. Saberi, and A. Zockaie. Urban network gridlock: Theory,
characteristics, and dynamics. Transportation Research Part C: Emerging Technologies, Vol.
36, 2013, pp. 480-497.
[26] Saberi, M., A. Zockaie, and H. Mahmassani. Network capacity, traffic instability, and
adaptive driving: Findings from simulated network experiments.In The 1st European Symposium
on Quantitative Methods in Transportation Systems, Lausanne, Switzerland, 2012.
[27] Saberi, M., and H. S. Mahmassani. Empirical characterization and interpretation of
hysteresis and capacity drop phenomena in freeway networks. Transportation Research Record:
Journal of the Transportation Research Board, Transportation Research Board of the National
Academies, Washington, DC, 2013.
[28] Zhang, L., T. M. Garoni, and J. de Gier. A comparative study of Macroscopic Fundamental
Diagrams of arterial road networks governed by adaptive traffic signal systems. Transportation
Research Part B: Methodological, Vol. 49, 2013, pp. 1-23.
[29] Gayah, V. V., X. Gao, and A. S. Nagle. On the impacts of locally adaptive signal control on
urban network stability and the Macroscopic Fundamental Diagram. Transportation Research
Part B: Methodological, Vol. 70, No. 1, 2014, pp. 255-268.
[30] Knoop, V. L., D. De Jong, and S. Hoogendoorn. The influence of the road layout on the
Network Fundamental Diagram.In Transportation Research Board 93rd Annual Meeting, 2014.
[31] Muhlich, N., V. V. Gayah, and M. Menendez. An Examination of MFD Hysteresis Patterns
for Hierarchical Urban Street Networks Using Micro-Simulation. Transportation Research
Record: Journal of the Transportation Research Board, Vol. 2491, 2015, pp. 117-126.
[32] Ortigosa, J., M. Menendez, and V. V. Gayah. Analysis of Network Exit Functions for
different urban grid network configurations. Transportation Research Record: Journal of the
Transportation Research Board, in press, 2015.
[33] Daganzo, C. F., and N. Geroliminis. An analytical approximation for the macroscopic
fundamental diagram of urban traffic. Transportation Research Part B: Methodological, Vol. 42,
No. 9, 2008, pp. 771-781.
[34] Leclercq, L., and N. Geroliminis. Estimating MFDs in simple networks with route choice.
Transportation Research Part B: Methodological, Vol. 57, 2013, pp. 468-484.
[35] Geroliminis, N., and B. Boyacı. The effect of variability of urban systems characteristics in
the network capacity. Transportation Research Part B: Methodological, Vol. 46, No. 10, 2012,
pp. 1607-1623.
[36] Newell, G. Blocking effects for synchronized signals.In Proc., 8th International Symp. on
Transportation and Traffic Theory, University of Toronto Press, Toronto, Canada, 1981.
Girault et al. 20
[37] Banks, J. H. Introduction to transportation engineering. 2002.
[38] Gartner, N. H., and C. Stamatiadis. Arterial-based control of traffic flow in urban grid
networks. Mathematical and computer modelling, Vol. 35, No. 5, 2002, pp. 657-671.
[39] Gartner, N. H., S. F. Assman, F. Lasaga, and D. L. Hou. A multi-band approach to arterial
traffic signal optimization. Transportation Research Part B: Methodological, Vol. 25, No. 1,
1991, pp. 55-74.
[40] Edie, L. C. Discussion of traffic stream measurements and definitions.In 2nd International
Symposium on the Theory of Traffic Flow, 1965. pp. 139-154.
