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.

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.

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)

Traffic databases LIMITED DATA FOR URBAN STREETS

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

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

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).

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

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

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

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

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.

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

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

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

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

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. 2001 data)

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

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

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

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

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.

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.

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.

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.

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.

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

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

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)

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

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

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

Dynamic Partitioning Shenzhen case study 20000 taxis (25M points/day) 9000 links 12M population Ongoing Ji and Geroliminis (2013) – Ongoing

Speed profile evolution Link Speed Histogram

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

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

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

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

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

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

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)

Gridlock Simulation WITH CONTROL BEFORE CONTROL R2 R1

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

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

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

Multimodal multi-reservoir system Buses Taxis Cars Taxis Cars

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

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

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

DISCUSSION