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Automated highway System with Traffic Density Control
Dr. Abdul Salam. A. AlKhazraji,
Fac. of Education – University of Garmian, Kurdistan Region – Iraq.
On the macroscopic level a roadway controller calculates the desired speed Commands to be followed by vehicles in each section of the freeway lanes in order to achieve desired traffic density distribution that lead to optimum traffic flow conditions.
In this paper we deign, analyze & simulate a roadway Controller for an automated highway that achieves desired traffic densities along the lane.
A macroscopic traffic flow model that is modified for (AHS) operation is used for control design & analysis.
We show that the proposed roadway controlled guarantees exponential convergence of the traffic density at each section to the desired density. Simulation results are used to illustrate the effectiveness of the proposed controller & the significant benefits (AHS) may bring to traffic flow.
Keywords: Macroscopic traffic flow: roadway controller: traffic flow density control: automated highway systems.
1. Traffic Flow Model
The analogy between traffic flow and fluid flow formed the basis for the first traffic flow model.
Fig. 1.
Consider a single freeway lane, which is subdivided into N sections with lengths , as shown in Fig. 1.
The space and timediscretized traffic flow model for a segment of the lane involves the following variables:
density in section i at time nT (in vehicles per kilometer per lane), where
space mean speed of vehicles in section I, at time nT (
traffic volume leaving section i, entering section at time nT (in vehicles per hour);
onramp traffic volume for section i (in vehicles per hour);
offramp traffic volume for section i (in vehicles per hour):
length of ith section (in km);
T timediscretization step size (in h).
The modified freeway traffic flow model given in Karaaslan et al. (1990) is in the following form:

..……….. 

………….. 
{

……….. 
Where
,
Here and are positive constants with , and k_{jam} is the maximum possible density. The variable in (3) represents the densitydependent equilibrium speed. In a manual operating environment with homogeneous traffic conditions. this relationship has been characterized in Papageor. giou (1989) and Papageorgiou ei al (1990a, b) as

………….. 
where and arc realvalued parameters, and V_{t}, is the free speed, which can be estimated from traffic data.
The term in (3) under manual operation depends on the downstream density, and can be expressed as

……………………… 
where the positive constant K is introduced to prevent abnormal growth of the velocity for section i when its density is very low,
Typical parameter values associated with the above model are given in Table 1.
2. Boundary conditions
We assume that the traffic flow rate entering section 1 during the time period nT and is and the mean speed of the traffic entering section 1 is equal to the mean speed of section In addition, we also assume that the mean speed and traffic density of the traffic exiting section N + 1 are equal to those of section Hence the boundary conditions for the entrance and exit can be summarized as follows:

…………………(6) 

…………………(7) 

………………….(8) 

…………………..(9) 
The physical meaning of each term of (3) that influences the mean speed of a section can be interpreted as follows (Papageorgiou, 1983, Karaaslan et aL, 1990).
Thc second term, is the relaxation term, which accounts for the evolution of the mean speed towards its density dependent equilibrium :speed with a time constant The dependence of on the density is influenced by the environment in which the traffic flow is operating For manual operation_{,} this relationship is governed by (4), as reported in Papageorgiou (1989) and Papageorgiou et al (1990 a.b). For an AHS operating under homogeneous heavy traffic conditions, the adopted safety policy for vehicles defines this relationship. For instance, if the desired safety distance between two vehicles Sa is made to depend on the equilibrium velocity
then the density equilibrium speed relationship can he characterized by

………………… 
Where is the inverse function of satisfies the equation If the constanttime headway policy is adopted for selecting the inter vehicle spacing then.
where h and c are constants. It follows that
The third term
in (3) is the convection term. It represents the influence of the incoming traffic on the mean speed evolution in segment i.
The last term,  in (3) is the anticipation term. For manual driving, human drivers in general increase or decrease vehicle speed, depending on the traffic density downstream. In other words, for manual driving, the anticipation term reflects the effect of downstream traffic density on the mean speed evolution in section i at sampling time nT. For instance, if the density downstream is lower. this term reflects the tendency of human drivers to increase vehicle speed. More specifically, the anticipation term represents the microscopic traffic dynamics that describes the interaction between vehicles or vehicles' response to the down stream's traffic conditions: For AHS, the anticipation term no longer represents the tendency of the human driver's response to the down stream's traffic conditions. Instead, It represents the microscopic traffic dynamics governed by the AHS control system.
Table 1.
Parameters associated with the traffic model.












93.1 
110 
1.86 
4.05 
0.95 
40 
4 
12 
6 
120 
35 
20.5s 
Km/h 
Vehicles Km^{1} lane^{1} 



Vehicles Km^{1} lane^{1} 
Vehicles Km^{1} lane^{1} 
Km^{2} lanc^{1} 
Km^{2} lanc^{1} 
Vehicles Km^{1} lane^{1} 
Vehicles Km^{1} lane^{1} 

The complete traffic model under manual operation is described by (1)(4), with w_{(n)} defined in (5)‑
The dynamics of traffic flow in AHS are still described by (1) (3), but the adopted safety policy that defines the equilibrium speeddensity relationship (10) replaces (4). Furthermore, the designed control law u(n) replaces the last term in (3). The complete behavior of traffic flow in AIIS is governed by the following equations:

………. 

………. 

………. 


Here are positive constants, with is the inverse function of the adopted safety policy .
3. Problem Statement
Traffic congestion in urban freeways is caused by ad hoc velocities and headway's that humans choose when they operate their vehicles. Therefore any strategy that hopes to reduce congestion needs to remove this human subjective element and replace it with a method that directly controls the density and speed of vehicles by
prescribing speed commands to vehicles in each section of the highway.
Assume that the roadway has the capability of measuring mean speeds and traffic densities at each section of a lane. The goal of traffic management is to assess the state of the traffic and provide appropriate speed commands to the vehicles at various sections of the lane in order to maintain a desired traffic density profile that under the current traffic conditions corresponds to some optimum traffic flow situation. A roadway controller can he designed to perform this task. The speed commands should be generated so that the desired traffic flow rate can be achieved and the density distribution along the lane leads to a homogeneous traffic flow.
Consider a lane subdivided into N sections with lengths L,, i = 1, ... N. as shown in Fig. 1. The traffic now rate entering section 1 at sampling time nT is q_{o}(n) vehicles per hour. The desired traffic density for section f of a single lane is assumed to be k_{d}(n).
Our objective is to choose a proper value of u,(n) for section r such that the traffic density of section i converges to the desired traffic density kw(n) exponentially fast, i.e k_{1}(n)+k_{d} (n) as .
4. A Roadway Traffic Density Controller
In this section, we propose a microscope roadway traffic density controller for AHS. Our design uses integrator back stepping to realize the control law needed to track a desired density profile The following general lemma Is used ,n
the design and analysis of the proposed roadway controller.
Lemma 1. Consider the following discretelime system:
where c is a constant and exponentially implies z(n).0 exponentially.
Proof. This is trivial and is omitted.
The main idea of the controller design is to apply back stepping and use Lemma 1 over and over again. That is, we treat v_{1}(n) as a free variable in (12) to control the density k_{1}(n + 1) at the next time point. Then the desired control action of u_{1}(n) is achieved by designing the control input v_{1}(n) utilizing (13).
The control design consists of three steps.
Step 1. We begin out design by defining the tracking error for section f as
Then, with (12), it follows that

…… 

…… 
Where
From Lemma 1_{,} we have and _{ }as . The goal of the next step is to choose the control input that guarantees _{ }as
Step 2. From the definition of and (II), we have

……. 
To simplify the notation, we define

……. 

…….. 

…….. 

……. 
We have from (12) and (15) that
Therefore. the quantities are available at sampling time nT. To stress this fact, we define

……. 
Substituting out definitions (18)(21) into (17). we obtain the compact form

……. 
Then, with (22), it follows that

…… 
We now consider the dynamics of this equation for the entrance . exit and intermediate sections of the freeway lane. Let us first define

……. 
Than, using (13) and (25) , we have

…….. 
Case for intermediate freeway sections. Using (24) and (26). we have
…..…… (27)
Where
…… (28)
Therefore is we choose the control signal to satisfy


And choose application of Lemma 1 to (27), we have
Case entrance. Using the boundary condition (7) in (24). we have


Substituting for a_{1}(n) and b_{1}(n) from (22). (18) and (19) in this equation and using the boundary condition (6), we have
...
Where

………… 
Therefore if we choose the control u,(n) to satisfy


then, by applying Lemma 1 to (31), we have
Case for freeway exit. Using the boundary condition (9) in (24). we have


Substituting for bn(n) and _{cn(n) }from (22), and (19) and (20) in this equation and using the boundary condition (8), we have

(35) 
Where

(36) 
Therefore is we choose the control to satisfy

(37) 
then, by applying Lemma 1 to (35), we have
Step 3. To obtain the control law u,(n), i = 1, 2, . N from (29), (33) and (37), we need to solve the algebraic equation

(38) 
Where

(39) 


(40) 


and
We define two constants , and where

(41) 

(42) 
Substituting (41) and (42) into the tridiagonal matrix equation (38) yields

(43) 
Where 
(44) 
The uniqueness of the solution of (43) depends on the nonsingularity of P(n). which in turn can be guaranteed if
for some constant.
To ensure satisfaction of this condition, we make the following assumption for each section of the freeway lane at any sampling time.
Assumption 2. (Traffic flow controllability). There exists a small positive constant such that , n,
Theorem 3 Assume that the traffic flow controllability stated in Assumption S is satisfied for each section at any sampling time. Leta be a positive constant defined as in Table I. Let P(n), U(n) and E(n) be as defined in (44) in (40). Then there exist control inputs u,(n) satisfying

(45) 
that drive the traffic density k,(n) for section i,i = 1, 2, .... N to the desired traffic density k_{4}(n) exponentially fast.
Proof. We showed in Step 3 that the desired control taw must satisfy the matrix equation (45). This provides the solution of the algebraic equations (29), (33) and (37) in Step 2. Application of Lemma 1 along with this solution to (27). (31) and (35) yields
as The use of the same lemma with (16) ensures that Hence we have exponentially.
Theorem 3 provides the control strategy needed for tracking a desired density profile. According to this theorem and (13), if u,(n) is chosen as (45), the average velocity in each section i of the lane at sampling time (n + 1)T will be

(46) 
and the traffic density at section t converges to the desired traf^{f}ic density k_{4} exponentially.
Consider a long segment of freeway, which is divided into 12 sections. The length of each section is 500 at. The initial traffic volume entering section 1 is assumed to be 1500 vehicles per hour. The initial density and mean speed of each section are as shown in Table 2.
Four cases are considered. In the first, shown in Figs 2 and 3, no feedback from the roadway is applied, and therefore u,(n) is replaced with the corresponding term in (3). From these figures, we see the propagation of congestion upstream due to the initial traffic congestion in sections 68, which eventually causes a traffic jam.
Table 2.
Initial densities and velocities of single lane freeway.
Section 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
Initial densities (vehicles km1lane1) 
18 
18 
18 
18 
18 
52 
52 
52 
18 
18 
18 
18 
Initial velocity (k mh1) 
81 
81 
81 
81 
81 
29 
29 
29 
81 
81 
81 
81 
Fig. 2. Density profile without control.
In the second and third cases, we use the proposed controller to achieve desired traffic densities of 23 and 35 vehicles per kilometer respectively. The simulation results shown in Figs 49 demonstrate that the initial congested conditions are quickly dampened out by the proposed controller, and the traffic flow is regulated to achieve the desired traffic densities.
In the fourth case, we assume that the input traffic flow rate in section 1 increases exponentially from 1503 to 2000 vehicles per hour as shown in Fig. 10. We set the desired traffic density for this cast to 23 vehicles per kilometer. The simulation results shown in Fig. 11 show that the desired traffic density is achieved exponentially with the proposed roadway controller.
6. Conclusions
In this paper, we have considered the use of feedback for controlling traffic flow on the macroscopic level in an AHS environment. We have considered the design and analysis of such a roadway controller using a macroscopic traffic.
Fig. 3. Velocity profile without control. Fig. 4. Density profile desired density
is 23 vehicles per kilometer.
Fig. 5. Velocity profile desired density is 23 Fig. 6. Density in each section desired
vehicles per kilometer. density is 23 vehicles per kilometer.
Fig. 7. Velocity in each section desired density Fig 8. Velocity profile desired density
is 23 vehicles per Kilometer. is 23 vehicles per kilometer.
Fig. 9. Density profile desired density Fig. 10. Density in each section desired density
is 23 vehicles per kilometer. is 23 vehicles per kilometer.
Fig. 11. Increasing entrance flow rate.
Flow model that is modified for AHS. The proposed controller has been shown to guarantee exponential convergence of the traffic density to the desired one. Simulations have been used to demonstrate that an initial density disturbance can cause instability, which manifests itself as congestion in the absence of feedback control. The use of the proposed roadway controller demonstrates that such instability can be counteracted and congestion can be avoided.
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