1. Macroscopic modeling of traffic in congested cities
Nikolas Geroliminis
Urban Transport Systems Laboratory
This talk will be more about physics than mathematics…
I hope to show both with simulation models and empirical evidence that some properties of traffic flow observed at the single-road level, such as the relationship between average speed and average density, scale up in the aggregate to large and complex networks. can be analyzed in the aggregate with little data.
These findings are not just of mathematical interest. They can impact the real-world because they unveil simple control strategies to improve an entire city’s auto-mobility.

2. State of the art Two options to relieve traffic congestion in cities
The traditional approach of building new infrastructure
DISADVANTAGES: economic factors, space and time limitations
Active Traffic Management (ATM) and Intelligent Transportation Systems (ITS)
Increase vehicle throughput, logistics operations and safety through the use of Integrated Systems with New Technologies and advanced sensors, by continuously observing the level of congestion and make propositions and decisions in real-time.

3. Develop travel time field and route choice information
MONITORING
Heterogeneity
Spatial and Temporal
Congestion Level
Topology
Modes of transport
Sensing equipment
Sparse multi-sensor data
Develop travel time field and route choice information
Usage of space (mixed traffic lanes, individual bus lanes, different conflicts for different topology and allocation of urban space)
GPS, camera, loop detectors in different penetration rates (% of vehicles or roads with sensors)
Pictures provided by Armando Bazzani (Un. of Bologna)

4. Traffic databases
LIMITED DATA FOR URBAN STREETS

5. Some ideas about modeling
Disaggregate Models
Data intensive
Unreliable Outputs
Aggregate Model
Observable
Reproducible Outputs
Ideal Gas Law
Disaggregate models that describe the city and its transport activity in great detail, and aggregate models that treat vehicles and people as masses.
Disaggregate models are appealing because they describe behavior (at least in theory) very precisely. But they require mountains of data that are difficult to obtain;
and the models cannot be expected to produce reliable outputs.
On the other hand, if you model transport in an aggregate way, like a water basin (a reservoir) where n is the number of vehicles in the system, this kind of model includes inputs and outputs that are more readily observable. Therefore these kinds of models can be verified and their predictions trusted.
For example, we may not be able to know the exact position or speed of each of the particles of a gas, but the state of an amount of gas is determined by its pressure, volume, and temperature which are all observable quantities.
As we will see, at the aggregate level, we can develop physically realistic models of urban congestion.
PV=nRT
DEVELOP PHYSICALLY REALISTIC MODELS OF URBAN CONGESTION

6. The promise of aggregate models
Move from PREDICTION to OBSERVATION
Develop physically realistic models of urban congestion
ROBUST APPROACH
As an alternative to the disaggregated models (that require a large amount of information, many times unobsevable) we propose to look for robust solutions that can be evaluated with robust models… i.e., observation-based models that circumvent the fragility problems traditional models.
In this proposed approach, monitoring replaces prediction, and the system is repeatedly modified based on observations.
To replace forecasting with measurement , Information Technology is on our side. This technology allows us to observe and quantify more and more things in real time.
PROPOSE→ MONITOR→MODIFY
PV=nRT
Information Technology
“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”.
JOHN VON NEUMANN

7. WHY MACRO?
Humans make choices in terms of routes, destinations and driving behavior (unpredictability)
Not a clear distinction between free-flow and congested traffic states. Empirical analysis of spatio-temporal congestion patterns has revealed additional complexity of traffic states (e.g. Helbing’s group work)
Need for real-time hierarchical traffic management schemes (decentralized control might not work for heavily congested systems)
Urban regions approximately exhibit a “Macroscopic Fundamental Diagram”(MFD) relating the number of vehicles to space-mean speed (or flow)
Robust linear relation between the region’s average flow and its total outflow (rate vehicles reach their destinations)
MFD is a property of the network infrastructure and control and not of the demand (time-dependent origin-destination tables are difficult).

8. Macroscopic modeling of traffic in congested cities
A Macroscopic Fundamental Diagram (MFD) for city traffic
Existence (Simulation evidence)
Existence (Empirical observations)
Description of dynamics
Perimeter control

9. Literature Review Thomson Herman & Ardekani Zahavi Godfrey Daganzo
Monotonic relationship between average speed and flow in UK
Herman & Ardekani
Field validation of two-fluid theory
Zahavi
relationship between traffic intensity, road density and speed
Godfrey
First MFD
1969
Daganzo
Gridlock Model
INITIAL MACROSCOPIC MODELS
TWO-FLUID MODEL
1966
1967
1968
1972
1979
1984
1987
2007
There are a number of interesting instances of macroscopic modeling of traffic in cities mostly in the 60’s and 70’s. Most of these works validated relationship between average speed and total flow.
In 80’s Herman and Prigogine developed the two-fluid model that has been tested in many studies and relates the average speed with the fraction of moving vehicles
Wardrop
effect of road geometry and traffic control on average speed
Mahmassani et al.
investigation through simulation
of network level relationships
Smeed
road capacity as a function of town area and density of roads
Herman & Prigogine
average speed is related to the
fraction of moving vehicles

10. Literature Review LIMITATIONS For the initial models:
1970
For the initial models:
no description of crowded conditions
The Gridlock Model:
description of the rush hour dynamically using an MFD
development of perimeter control
FOR ONE-RESERVOIR CITIES
TODAY
For the two-fluid model:
no description of the rush hour dynamically
no detailed sensitivity analysis for different O-Ds
no development of control strategies
The limitations of these models are models:
The initial models cannot describe crowded conditions. They propose speed-flow curves that are monotonically decreasing. But what we know today is that you can observe small flows for high and small speeds as we can see in this movie.
The two-fluid model captures this limitation but is not able to describe the dynamics of the rush hour. Also , it does not contain any sensitivity analysis for diferrent demand profiles and control strategies to improve crowded conditions.
Gridlock model described the rush hour and developed control strategies for the case of single reservoir cities. But this model never tested for its existence.
They did not demonstrate that an invariant MFD could dynamically arise in the real world.
Daganzo
Gridlock Model
INITIAL MACROSCOPIC MODELS
TWO-FLUID MODEL
1966
1967
1968
1972
1979
1984
1987
2007

11. Fundamental Diagram (FD) for a link i
Accumulation : ni (vehs)
Travel Production : Pi (veh-km/hr) - VKT
Output-Trip completion rate (vehs/hr) Pi ni
Qi(ni)
3 Regimes
I : Undersaturated
II : Efficient
III : Oversaturated
Growing queues from the downstream link block the arrivals

12. Theory: Generalization to networks
AGGREGATE BEHAVIOR =
SCALED UP VERSION OF LINK BEHAVIOR
VKT
Accumulation
Our conjecture is that if a neighborhood of a city (reservoir) is (roughly) homogeneously loaded and congestion is (roughly) evenly distributed over the network, then its production P which, is the sum of the measures for individual links and can be expressed as a function of the total accumulation of the network independently of the disaggregate link data.

13. Key findings Output Production Output Production Accumulation
1970
Output
Production L
Output
(Number of trips ending/u.t.)
Production
(Veh-km traveled/u.t)
While the 1970 models propose curves between travel production and accumulation similar to this, our understanding is that this diagram has 3 different regimes.
VKT is low when either few vehicles are there (the case shown in green) or the system is jammed (the case shown in red).
Note too that there is a “sweet-spot” accumulation when output is greatest (the middle case, shown in yellow).
For being able to describe the real dynamics of a city street network the reservoir outputs are needed. There is a reproducible relation between the number of vehicles in the reservoir at any time (the accumulation) and the rate at which vehicles exit the system (the reservoir output) similar as the one between Vh-km traveled and accumulation.
Accumulation
(Number of vehicles in the network)
An MFD exists
Trip completions / Network flow ≃ Constant

14. Macroscopic Fundamental Diagram (MFD)
Output
Vehicle Accumulation
Output
This is a 4-hour simulation of the San Francisco business district with higher demand than in reality, showing how accumulation grows and how this affects production and outflow in the predicted way. The right side is the street network. Green and red dots are the traffic signals and the moving dots are vehicles.
The left side will show the accumulation and output values that arise in our simulation as time passes, and how the observable Vh-km and unobservable output are related. Note how these values closely follow our predicted curve. Note again the high outflow when accumulation is in the range that maximizes outflow; and how outflow drops toward zero when accumulation is allowed to exceed the ideal amount.
This suggests that by monitoring and managing accumulations we can predict and control outputs, and in this way improve mobility.
The multi-colored dots that have appeared are simulations of additional days with slightly different demands – and note how these new dots still follow the same curve.
This means that this curve can be used to make reliable control decisions. For example, we might try to discourage vehicles from entering a neighborhood that is too crowded by (for example) timing the signals, or congestion pricing.
+Different demand
Production

15. MFD – Sensitivity to Demand
DESTINATIONS
PER NODE
ORIGINS
DESTINATIONS
PER NODE
ORIGINS
Output
Vehicle Accumulation
MFD is a property of the infrastructure INDEPENDENT OF O-D TABLES

16. INPUT OUTPUT ESTIMATED OUTPUT ACCUMULATION
MFD - Dynamics n
e =G(n) q
G(n)
RUN 1
We applied this differential equation to two different runs of the SF network with a time-dependent O/D table … we also ran a micro-simulation with the same data as a benchmark for comparison… and here is what we found.
The curves you see give the cumulative number of vehicles to have been counted as a function of time.
The last two curves really fit quite closely… they are almost perfectly superimposed
Predictions when traffic is in the RED regime are more difficult because the system is chaotic and not in steady state conditions. But a control strategy should try to avoid these states if they arise.
RUN 2
INPUT OUTPUT ESTIMATED OUTPUT ACCUMULATION
Geroliminis and Daganzo, 2007
Daganzo, 2007

17. Real World Experiment: Site Description
Fixed sensors
500 ultrasonic detectors
Occupancy and Counts per 5min
Mobile sensors
140 taxis with GPS
Time and position
Other relevant data
(stops, hazard lights, blinkers etc)
Geometric data
Road maps
(detector locations, link lengths, intersection control, etc.)
10 km2
(Dec data)

18. Single Detectors flow occupancy
Let us look at single detectors data first. This is a scatter plot of flow vs. occupancy with a time slice of 5 min for a detectors for a whole weekday.. Note the scatter, especially when flows are maxima
This pattern is typical and here is another detector… POP
Note that the scatter is similar.
But, what would this scatter persist if one aggregated data from all the detectors?
occupancy

19. Aggregated Demand Average Occupancy by time-of-day
To test our hypothesis with very different demands data were aggregated for two different days: a weekday and a weekend day.
These figures show the time-series of average flows and occupancies that were observed on these two days. On the left we have occupancies and on the right flows. Black is used for the weekday and Grey for the weekend day. You can see that the demand was quite different on both days and also within each day.
The substantial variations within and across days suggest that the demand rates and origin destination (O-D) tables indeed varied considerably during our observations.
Accordingly we have used eight different time periods to classify our data.
Let us now see whether these occupancies and flows are related in some way
Average Occupancy by time-of-day
Average Flow by time-of-day

20. All Detectors Average flow Average occupancy
Here is what we found when we plotted all the data from both days on the same diagram.
This result stunned us for its orderliness. Particularly since it includes data from all eight time periods. They are marked by different symbols.
Note: (1) there is no evidence of hysteresis; (2) the sweet spot, reached for an average network occupancy of about 0.3, is the same on all the periods; (3) there indeed appears to be a MFD.
Average occupancy

21. All Detectors No dependence on O-Ds No traffic hysteresis 1970 speed
TODAY
Another way of expressing the same data is by plotting the average network speed vs. the average occupancy and here is the highly ordered pattern we found.
this MFD only describes that part of Yokohama’s network which is covered by detectors.
occupancy
flow
No dependence on O-Ds
No traffic hysteresis

22. Fusing taxi and detector data
An MFD exists on the part of the network covered by detectors. What about the whole network?
1. Filter for passenger-carrying taxis (i.e. full)
2. Estimation of accumulation and speed
3. Results
To test this hypothesis we used the taxi data because taxis cover the complete network; including the very minor streets.
But, taxis move quite differently than cars when looking for passengers and they stop more frequently and not necessarily because of congestion.

23. Filters to determine full vs. empty taxis
1. Identify passenger moves (boarding or alighting) if:
hazard lights are ON or
parking brake is used or
left blinker is ON and taxi stops > 45 sec or
speed < 3 km/hr for >60sec
2. Identify full trips:
trip duration > 5 min and length > 1.5 km and
trip distance < 2 × “Euclidean distance”
The filter first determined with some simple criteria whether a taxi stop involves a passenger boarding or alighting. The basic idea is that stops related to passenger moves are longer.
We then determined with other criteria whether trips between these types of stops were full (i.e. passenger-carrying) or empty. The idea now is that taxi moves are more circuitous when looking for passengers.

24. Illustration of Filter Results
The filter seems to produce reasonable results….
The slide shows by means of white lines the complete set of taxi routes for one week, which accurately reproduce the area’s map.
The perimeter of A is shown by a dashed red line.
The figure also shows the trajectory of a taxi for 3 hours. The large symbols are stops for loading or unloading a passenger. Smaller symbols show the position of the taxi every 30 s. When the taxi is empty these symbols are close together and the route is cicuituous. When the taxi is carying a passenger, the simbols are more widely spaced and the route is more direct.
Thus, we make the hypothesis that they are representative of car trips as we had assumed.

25. Estimation of accumulation and speed
Conjecture: Passenger carrying taxis use the same parts of the network as cars
Then:
Here is what we did.
Our goal is estimating the number of cars in Yokohama’s center at regular time intervals, as well as their average speed at those times. And then to construct a scatter plot of accumulation vs. speed.
Remember too that the cars in Yokohama’s center (the grey dots on the screen) are invisible, but the taxicabs (black squares with circles) are tracked.
Our method is based on the conjecture that taxi passengers make similar trips as cars, which we just showed that looks like a reasonable assumption.
we can estimate the average network speed by the average speed of passenger-carrying taxis.
And we can also estimate the number of cars if we knew the ratio of cars to passenger-carrying taxis in the network… phi.
But if passenger carrying taxis (squares with dots) behave as cars (dots) both would have the same probability of exiting the zone on streets with detectors – where both can be seen.
These exist are shown here in red. If passenger-carrying taxis and cars behave similarly one would expect the ratio of red dots over red squares to be the same as grey dots over black squares. And this is the basis for our estimation: the estimated number of cars is simply the observed number of passenger-carrying taxis, multiplied by an expansion factor which equals the ratio of cars to passenger-carrying taxis exiting the zone.

26. Full taxis = Cars
We also tested this hypothesis systematically in the only way possible given the available data.
This figure shows the ratio of inbound to outbound trips along the periphery of our zone. The dark line is the ratio obtained from detector data, which includes all cars; and the grey line (with greater fluctuations) the ratio for valid taxi trips.
Note that despite the fluctuations (which are large only during the night between 11:00 PM and 6:00 AM) the taxi data track the automobile data quite well.
This suggests that the filter was successful in extracting from the taxi data those trips that are representative of car trips.

27. Real World Experiment: Results
MFD is a property of the network independent of demand.

28. Trip Completion (vh/min)
Spillovers and MFD
Trip Completion (vh/min)
Spillovers (vhs)
Max
Actual
Spillovers (vhs)
Speed (km/hr)
Spillovers (vhs)
Accumulation (vhs)

29. Properties of a well-defined MFD
dr(t): pdf of individual detectors’ density in region r
Q(t) and O(t): Average network flow and density
Variance much higher than binomial’s
WHY?
Correlation of link density (propagation)

30. An example of a “bad” MFD
Strong hysteresis phenomena in freeway MFDs
EXPLANATION
Different distribution of congestion (onset vs. offset)
Synchronized Phase Transitions in individual locations
Average flow
Freeway network of Minneapolis (USA)
Average occupancy
Average flow
Density Variance
Average occupancy

31. Spatial Variability and Network Capacity
A key variable to understand well defined MFDs is spatial variance. Let’s see how this variance can affect the network capacity.
Networks have high degree of uncertainty in the performance
30x30 link network with mesoscopic simulation
Graph 2: MFDs for the same value of spatial variance
These results tell us that the network flow for heterogeneous networks depends on the (i) number of vehicles in the network and (ii) spatial distribution of these vehicles
Mazloumian, Geroliminis and Helbing (2010) – Ph.Trans.Roy. Soc. A

32. Congestion Spreading – Intro to Partitioning
Ji and Geroliminis (2012) – Trans. Res. Part B

33. Dynamic Partitioning Shenzhen case study
20000 taxis (25M points/day) links M population
Ongoing
Ji and Geroliminis (2013) – Ongoing

34. Speed profile evolution
Link Speed Histogram

35. Identify congestion propagation
Estimate link speeds from taxi data
6am - 8am, 15-min interval every 5 min
Identify and smooth congested links
Congested link speed <= 1/3 max speed
Spatial smoothing
If an uncongested link has more congested neighbors, this link becomes congested.
Estimate the largest connected components (CC)
breadth-first search
a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths

36. Congestion Propagation
Small number of critical pockets of congestion
Dynamic partitioning is feasible with CC

37. Management Options: Control
Without affecting
# trips per mode
By affecting
# trips per mode
With estimates of real-time accumulations one could develop many policies to keep accumulations in their sweet-spots so as to enhance mobility.
Modifying traffic signal timings would be one way of doing this without affecting the number of trips per mode.
But we could also enact strategies that would change the number of trips per mode in favor of the more sustainable modes (e.g., shifting trips from cars to buses). We know that pricing strategies (for example for parking or peak hour tolls) can do this; So significant shifts in modal share should be accompanied by shifts in how space is allocated to modes.
Signal Control
Urban Space
Allocation
Pricing
Parking
Change
Road Share
Change
Mode Share

38. Macroscopic Perimeter control
VKT1
Neighborhood 1 B R1 R2 A A D
Neighborhood 2 n1
VKT2 R1 R2 B C C D n2

39. Multi-region cities Partitioning + Control, BUT…j k
Control will be discussed later

40. Effect of Perimeter Control
No Control
With Control
Trips Ended
Time
Trips Ended
This animation shows how with this kind of control; specifically, by using pre-timed traffic signals to restrict vehicles from entering the reservoir at too great a rate we prevent accumulation from growing beyond its sweet spot. As a result, total output is enhanced.
The curves in the middle give the actual number of vehicles that have finished their trips in each scenario.
the blue curve on the bottom describes the scenario with control while the red curve on top is this without control.
The simulation shows that the blue curve (describing control) reaches a greater height; that is, more destinations are reached in the time of the simulation. Thus, control increases accessibility.
If we monitor the system and vary the control in response to changes in accumulation, we could obtain a blue curve that would increase even more rapidly. In other words, control can put us in the sweet spot more effectively if the system can be monitored.
Let us now talk about control a little more.
Time

41. Dynamics of a 2-reservoir system
State Variables
Boundary Capacity
Dynamic Equations
Control
output ni
Gi(ni)
One can model a city as a single or multi-reservoir system depending on the geometry, the demand patterns and the distribution of trip destinations among the city. The requirement for homogeneity in traffic loads should determine the number of required reservoirs. Consider a city partitioned in two reservoirs. This could be the case where the internal reservoir attracts most of the trips during the morning commute and the external generates most of the trips at the same time period. Using conservation of mass we can develop the dynamic equations in this case.
The causality of this function is that sufficiently large accumulations restrict the input from the external reservoir.
= IN OUT
Boundary
capacity ni
Cji(ni)

42. Gridlock Simulation
WITH CONTROL
BEFORE CONTROL R2 R1

43. Simulation Numerical Results
# Trips ended
With Control
Before Control
in R1
89695
9947
in R2
90218
82090
Started from R1
15892
4037
Started from R2
164020
88000
Total
179913
92037 R2 R1
PARETO IMPROVEMENT

44. Multimodal networks
In urban networks, buses usually share the same network with the other vehicles.
Movement Conflicts in multi-modal urban traffic systems: Modeling Congestion and developing more sustainable cities
Bus stops affect the system like variable red signals in a single lane (instead of blocking all lanes).
Increasing bus frequency decreases the flow of vehicles but can increase the flow of passengers.
Bus stops effect like single lane red phase.
Performance Measures
Vehicle Hours Traveled
Vehicle Kilometers Traveled
Passenger Hours Traveled
Passenger Kilometers Traveled
Mobility (Accessibility)
Emissions (Environ. Impacts)
Costs (Users, Providers, etc.)
Road Space Used
Competing modes
Parking
Pax vs. veh throughput
MULTIMODAL CITIES

45. More people compete for limited urban space with different modes.
Mobility and transportation are two of the leading indicators of economic growth of a society. More people and modes compete for limited urban space to travel. We need to understand how this space is used and can it be managed to increase accesibility.
More people compete for limited urban space with different modes.
We need to understand:
How this space is used
How it can be managed to improve
Accessibility
Sustainability
Macroscopic methodology to model traffic with different modes
How throughput of passengers depends on system characteristics
How to allocate city space to different transportation modes

46. Multimodal multi-reservoir system
Buses
Taxis
Cars
Taxis Cars

47. Simulated data – Downtown SF
Network performance measures
Bus Density
Bus Density
Car Density
Car Density
VEHICULAR FLOW
PASSENGER FLOW
Simulated data – Downtown SF

48. Infrastructure not equally available throughout a city
Of course, modes can only be effectively separated if the road is sufficiently wide.
When devising a plan for allocating space for various modes, there will typically be some links that are too narrow to accommodate separation schemes (e.g. near bottlenecks). But queues will often spill-over to other (wider) links. In wider links, separation can produce benefits to reduce people-hours of delay; e.g. by providing a congestion bypass lane for buses.
Queues form at locations with limited capacity,
but spill-over to other locations 48

49. Need not provide special lanes everywhere
General Lanes
Congested
City Center
Example
Streets for only buses in a dense urban network
You may therefore deploy bus lanes in the same way.
A nice feature of city street networks is that they are dense.
-click-
While we would not devote an entire freeway to HOVs, we may devote an entire street to buses (the green street in the picture) if nearby parallel streets can be devoted to the other modes.
This is most effective if we can design the bus lines in the city to fully use the dedicated space without congesting it. One of the authors found from experience that Oxford Street in London, shown here, which serves 19 bus lines, tends to be congested.
These ideas shed only some light on the behavior of reservoirs (neighborhoods) with 2 modes. Our understanding of 2 mode reservoirs is not yet complete. Further empirical and theoretical work on the behavior of traffic with modes that are very different is needed.
Provide bypasses for more efficient modes
around much (if not all) congestion 49

50. DISCUSSION