04 - Microscopic traffic flow modeling - OMT

04 - Microscopic traffic flow modeling - OMT

Introduction

Microscopic traffic flow modeling falls under the scope of [[Car following theory]].

This kind of theories work well only for heavy traffic conditions.

Definitions

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2 vehicles follow each-other.

The vehicle in front is referred to as Leader and is vehicle n of all the vehicles on the road. The following vehicle is referred to as Follower and is vehicle n+1.

Some parameters:

Introduction

The objective of microscopic traffic flow modeling is trying to predict the trajectories of individual vehicles on space and time. The movement has 2 components:

In this chapter we will only look at longitudinal models.

Spacing in central for 2 concepts:

Safety vs flow

Spacing needs to ensure that drivers can react and adapt to new speeds of vehicles in front without colliding. In terms of safety, the larger the spacing, the safer. In contrast, a large spacing, at equivalent speed, causes a reduction in capacity of the infrastructure:

q=vs

In terms of capacity, it would be ideal to have high speed and small spacing.

On the road, spacing is left to the driver perception of risk. It's the drivers that evaluates and decides what is the appropriate trade-off for each situation to guarantee safety. In other kind of transportations, like trains, it's the infrastructure that tells the driver the spacing, or better, it forces the appropriate spacing.

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Automated driving

Automated driving promises to brake the trade-off [[#Safety vs flow]].

Autonomous vehicles can achieve very short reaction times. If we assume that all vehicles are able to communicate with each other, then the reaction time is barely the time it takes the information to travel from one veh to the other. The order of magnitude is 1ms. Also, because of communication, all veh are able to react together even without line of sight.

Therefore, Autonomous vehicles should be able to travel with very little spacing.

The diagram shown in [[#Safety vs flow]], with autonomous vehicles, could become more similar to the following:

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Notice how the spacing will be fairly constant, despite the speed, in heavy traffic, to a point where spacing is not dependent on speed anymore in light traffic.

Platooning

What [[#Automated driving]] is supposed to achieve is known as platooning.

platooning

Platooning is a condition where many vehicles travel as if they were one, connected as a train.

Car following models

Several microscopic models were proposed during the years. Here we will see some of them. In order to study them, it's important to first understand the following graph:

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From now on we will have to account for the vehicles length, hence, each vehicle is represented by a pair of trajectories (one for the front and one for the tail of each vehicle). All the main parameters are shown in the graph:

The following relations can be deduced:

sn+1(t)=ln+1+gn+1(t)sn+1(t)=hn+1(t)x˙n+1(t)

Now we are ready to see several models. Specifically, we will see:

Pipes car-following model (1953)

Pipes' was one of the first [[#Car following models]] ever proposed.

It comes from the [[California Driving Code]]. The code stated: "Drivers should leave a gap of one vehicle length for every 10mph of traveling speed."

This translates to:

pipes model (1953)

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According to the Pipes model, Drivers should leave a gap of one vehicle length for every 10mph of traveling speed:
sn+1(t)=ln+1+x˙n+1(t)10mphln+1

This model works quite well although:

This model is incredibly simple: in fact it has only 1 parameter: the follower length ln+1.

Forbes car-following model (1958)

In 1958, Forbes introduced [[Reaction time (traffic)]] (Δt) into the model.

It was proposed that, in order to avoid collisions, the time gap between 2 vehicles should be at least equal to the reaction time.

Therefore, the headway should be the reaction time Δt plus the time it takes the vehicle to cover one vehicle length ln+1x˙n+1(t).

hn+1(t)=Δt+ln+1x˙n+1(t)

Since the spacing is given by

sn+1(t)=hn+1(t)x˙n+1(t)

Forbes model, formulated in terms of spacing, then becomes:

sn+1(t)=(Δt+ln+1x˙n+1(t))x˙n+1(t)==ln+1+Δtx˙n+1
forbes model (1958)

sn+1(t)=ln+1+Δtx˙n+1

Like pipes' model, this is also linear. This time though, it is dependent on a parameter that is not fixed, meaning the model can be calibrated to better describe the specific traffic observed.

General motors car-following models (1958-1961)

In the late 50's, American company [[General Motors]] started developing some [[#Car following models]]
They developed several generations of a model. The models were calibrated using real world data.

General Motors car-following models fall under the spectrum of [[stimulus-response model]].

GM car following models

stimulus-response models

Stimulus-response models, in car-following theory, describe the acceleration of the follower as a function of the speed relative to the leader.
Response=f(stimuli,sensitivity)

Response

In longitudinal movement, the follower can only react in 2 ways to what the vehicle in front does:

The response is therefore measured in terms of follower acceleration:

x¨n+1(t+Δt)

Stimulus

Sensitivity

measures how attentive you are to traffic.


The general model is always in the form:

x¨n+1(t+Δt)=α(...)[x˙n(t)x˙n+1(t)]

Depending on the model generation then, α is expressed as different functions.

GM model Generations

1st generation - GM model

In the 1st generation of the #General motors car-following models (1958-1961) the sensitivity α is considered constant:

x¨n+1(t+Δt)=α[x˙n(t)x˙n+1(t)][ms2]=[1s][ms]

The acceleration of the follower is directly proportional to the relative speed between the leader and the follower.

α and Δt are considered constant parameters to calibrate.

When DM tried to calibrate the model, they realized that they weren't able to find a constant falue for α. This was instead possible for Δt.

The sensitivity parameter showed large variability depending on the driving conditions, suggesting that maybe this parameter was not a constant.

2nd generation - GM model

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GM observed that α had too much variation. So, they decided to use 2 different values of α

3rd generation - GM model

Since they didn't know at what point you would get a new value of α, GM decided to use a linear function to describe α:

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x¨n+1(t+Δt)=α0[xn(t)xn+1(t)][x˙n(t)x˙n+1(t)]

where α0 is the sensitivity coefficient and is measured in [ms].

So, α0 is a speed. It gives the linkage between microscopic and macroscopic traffic flow models.

Relation between 3rd gen micro model and Greenberg macro model

Integrating the 3rd generation model in time, we get (notice that the relative speed is the derivative of the spacing):

x˙n+1(t+Δt)=α0ln(xn(t)xn+1(t))+C

In terms of speed:

V=α0ln(Ck)

We can try to find the constant of integration C by the following initial conditions:

V=0k=kj

then:

0=α0ln(Ckj)C=kj

and we get:

V=α0ln(kjk)

which looks exactly like the 03 - Fundamentals of traffic flow modeling - OMT#Greenberg k-v model (1959) with α0=v0, which is the optimal speed.

4th generation - GM model

In the 4th generation, they proposed that α0 is not constant but dependent on the speed of the follower:

α0=αx˙n+1(t)

the model is then

x¨n+1(t+Δt)=αx˙n+1(t)[xn(t)xn+1(t)][x˙n(t)x˙n+1(t)]

where

This means that how attentive you are, not only depends on the spacing, but also on the traveling speed (more attentive the faster you go)

! We still have 2 parameters.

5th generation - GM model

2 parameters are added, n and l. They have no known physical meaning, but they're useful to hav more flexibility in calibrating the model.

x¨n+1(t+Δt)=α[x˙n+1(t)]n[xn(t)xn+1(t)]l[x˙n(t)x˙n+1(t)]

All the [[#General motors car-following models (1958-1961)]] are a particular case of the 5th gen model.

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Traffic stability

The kind of equations we setup in the [[#General motors car-following models (1958-1961)]] define an oscillatory movement.

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We can look at an instability parameter, found from the models:

C=αΔt

Notice that C is a-dimensional.

It can be shown that traffic is stable if:

C0.5

If C>0.5 you get unstable traffic: overreaction.

This condition is valid for asymptotic stability: it's not only a pair of veh, it's a long queue of vehicles.