traffic flow models using second-order ordinary differential equations
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Traffic Flow Models Using Second-Order Ordinary Differential Equations. Vane Petrosyan Mariam Balabanyan. Overview. What is traffic flow? Mathematical model of traffic flow Application of the model to investigate stopping distances - PowerPoint PPT PresentationTRANSCRIPT
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TRAFFIC FLOW MODELS USING SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Vane PetrosyanMariam Balabanyan
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Overview
• What is traffic flow?• Mathematical model of traffic flow• Application of the model to investigate stopping
distances– Solving an initial value problem using a linear, non-
homogenous, constant-coefficient, second-order differential equation
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What is traffic flow and why is it important?
Exploration of interactions between vehicles, drivers, and infrastructure in a general environment
Goal of researchers: Generate a useful model Promote a road network with efficient
movement of traffic and nominal traffic congestion interference
Three variables: speed, flow and concentration
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A Mathematical Approach
Microscopic models consider the behavior of individual vehicles and their motion in relation to each other
Car-following model that describes how driver behavior is influenced
011
21
2
xxdt
dx
dt
xd
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Achieving a simplified version
= sensitivity coefficient T = reaction time d = preferred separation between the two
vehicles = position of the lead car = position of the following car
0x
1x
))()(()(
1021
2
dtxtxdt
Ttxd
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Inclusion of the reaction time T makes it difficult to solve the equation (although it should be included in more realistic models)
We will consider the quick thinking driver version, where T=0
))()(()(
1021
2
dtxtxdt
txd
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Our preferred separation tends to depend on our speed
Short distance in slow moving traffic
Greater distance in faster moving traffic
Assume that the following driver’s preferred separation is dependent on her velocity
dt
tdxd
)(1
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Substituting d into the original model, yields the simplified version
))()(()(
1021
2
dtxtxdt
txd
))(
)()(()( 1
1021
2
dt
tdxtxtx
dt
txd
dt
tdxtxtx
dt
txd )()()(
)( 1102
12
)()(
)()(
01
121
2
txdt
tdxtx
dt
txd
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011
21
2
xxdt
dx
dt
xd = sensitivity coefficient
(’s value is directly proportional to the reaction of the following driver to the relative velocity between vehicles)
= preferred temporal separation (seconds)
= position of the lead vehicle
0x
So the model we will be using is…
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Application to investigate stopping distances
“Consider two cars driving along the road at constant velocity U m/s and separated by a distance D meters. At time t = 0 we assume the following car is at the origin t 0, D = . How does the following car respond?” (McCartney 591)
0x
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Becomes the following initial value problem
U
dt
dx
01 0)0( x
Dxdt
dx
dt
xd 1
121
2
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01
121
2
xdt
dx
dt
xd
stetx )(1
11
21
2
xdt
dx
dt
xd )()()(
2
2st
stst
edt
ed
dt
ed ststst esees 2
stess )( 2 steSince is never zero, we must have
)(0 2 ss
First we must find the general solution of the corresponding homogenous equation.
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From the characteristic polynomial, we can deduce the roots of the equation
So we are given 3 cases:
2
42
1
r
2
42
2
r
=4, two repeated roots
2>4, two real and distinct roots
2<4, two complex roots
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Case 1: 2=4
DAxp
DAtececx rtrt 21
rtrth tececx 21
rtrtrt rtececercx 221 rtrtrt tercercecrx 2
2212
Dxdt
dx
dt
xd 1
121
2
DDAtececrtececectercercecr rtrtrtrtrtrtrtrt )()()( 212212
2212
DDA 2
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D
D
D
DA
2
1A
1
A
1
21 Dtececx rtrt
Dtececx rtrt 21
Plug in the initial condition, to solve for the value of c1:
0)0( x
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00)0( 02
01 Dececx rr
0)0( 1 Dcx
Dc 1
Now use the initial condition given for x’(0)=U to solve for c2
U
dt
dx
01
rtrtrt rtececercx 221
UercecDrex rrr 02
02
0 00
UcDrx 20DrUc 2
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DteDrUDex rtrt )(
So substituting c1 and c2 into the equation yields:
this equation can be rewritten as
UteDe
DeDx tt
t
/2
/2/2 2
which can then be simplified to
tUD
DeDtxt
)2(
)(/2
1
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Case 2:
DtDtDUe
txt
)4(2
1cosh)4()4(
2
1sinh2
)4()( 222
2
)2/1(
1
42
• Has two real and distinct roots
• Solution includes hyperbolic sine and cosine function
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Case 3: 42
DtDtDUe
txt
)4(2
1cos)4()4(
2
1sin2
)4()( 222
2
)2/1(
1
•2<4, two complex roots
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What this means physically? By making case 3 equal to D we ultimately see
the following:
0= asin(x) - bcos(x)
0= atan(x) – b
tan(x)= b/a
So there are infinite times where tan(x) = b/a
DtDtDUe
txt
)4(2
1cos)4()4(
2
1sin2
)4()( 222
2
)2/1(
1
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Conclusion
The mathematical model will only be physically meaningful if we restrict parameter values
The driver may be quick thinking but the vehicle will infinitely collide
the stopping distance is proportional to the initial velocity of the driver