stability analysis of an extended intelligent driver model and its simulations under open boundary...

Physica A 419 (2015) 526–536

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Physica A

journal homepage: www.elsevier.com/locate/physa

Stability analysis of an extended intelligent driver model andits simulations under open boundary conditionZhipeng Li ∗, Wenzhong Li, Shangzhi Xu, Yeqing QianThe Key Laboratory of Embedded System and Service Computing supported by Ministry of Education, Tongji University, Shanghai,201804, China

h i g h l i g h t s

• We proposed an extended intelligent driver traffic flow model with power cooperation.• Linear stability analyses have been conducted.• The effect of power cooperation is examined theoretically as well as numerically.• We simulated the traffic flow model on a single lane under open boundary condition.

a r t i c l e i n f o

Article history:Received 15 September 2014Available online 18 October 2014

Keywords:Intelligent driver modelOpen boundary conditionPower cooperationTraffic flow stability

a b s t r a c t

This paper presents an extended intelligent driver traffic flowmodel, inwhich the power ofthe considered vehicle is strengthened in proportion to that of the immediately precedingvehicle. We analyze the stability against a small perturbation by use of the linear stabilitymethod for the proposed traffic flowmodel on a single laneunder openboundary condition,with the finding that the traffic flow stability can be improved by increasing the proportionof the direct power cooperation of the preceding vehicle. The participation of forwardpower cooperation can help to stabilize the traffic flow and suppress the traffic jams. Inaddition, the simulations under open boundary single lane are conducted to validate thecorrectness on theoretical deduction, which shows that numerical results in large-waveand short-wave stability are in good agreement with those of theoretical analysis.

1. Introduction

Due to the seriousness of traffic fumes and the great importance of efficient traffic for modern countries, the study oftraffic flow theory has been given considerable attention in the past several decades. Manymodeling approaches have beenapplied to describe the complicated dynamics of traffic flow in fields [1–3]. Traditionally, there are two types, microscopicand macroscopic models [4–12] are distinguished for the stability analysis. According to the viewpoint of microscopic ap-proaches, each vehicle is modeled by its own equation of motion. In contrast, macroscopic models represent traffic flow bya continuum approach needing several traffic variables like the spatial traffic density, average speed and traffic flux [13–18].

The intelligent driver model (IDM, for short) is one of the favorable car-following models [19–22], which is describedby a second-order differential equation and the motion of a vehicle depends on the velocity, the net distance gap and thevelocity difference to the leading vehicle. An increasing number of investigators with different backgrounds and points ofview devoted themselves to the study of traffic dynamics by use of the IDM model because of its strong advantages in

∗ Corresponding author.E-mail address: lizhipeng@tongji.edu.cn (Z. Li).

Z. Li et al. / Physica A 419 (2015) 526–536 527

simulation. The effects of some human elements (the finite reaction times, the estimation errors, the spatial anticipation,and so on) on traffic characteristics have been investigated to explore some inherent patterns existing in real traffic. Theimpacts of traffic stability have been proved to depend on the driver behaviors and different driving habits. In addition,manyimprovements in themotion equation of the original IDMmodel have been conducted for a variety of desired purposes, andthe string stability analysis is usually implemented in the IDM model for the purpose of verifying the availability of someimprovements [23–26].

In recent years, with the boom of information and communication technologies (ICT), the fantasy that driving tasks areshifted from the driver to the intelligent equipped vehicle, has become reality today by integrating the technologies of thesensors, the global positioning system and intelligent control. The rapid development of the connected vehicle has achievedthe real ‘‘Car Talk’’—vehicles that can communicatewith each other by using vehicle Dedicated Short Range Communications(DSRC, for short). Many scholars were engaged in the exploration to strengthen the stability of traffic flow by use of theavailability ofmotion information from the preceding vehicles on the same road, such as the headway, velocity and flux [27].Theoretical analysis results show that some improvements play a great role in avoiding the appearance of traffic densitywaves (go-and-stop waves). However, most of them were proposed to adjust the acceleration of the considered vehicleby indirectly integrating the acceleration of the vehicle ahead. Moreover, they have conducted the linear stability analysisand traffic simulations to their models under the periodical boundary condition. So it is inevitable that there exist someshortcomings describing the real traffic system in these models.

In viewof that, the task of this paper is to deliver amore direct approach of IDMmodel considering the power cooperationof the nearest vehicle in front. Then an extended model is proposed and is used for simulations, the linear stability analysiswill be done for the proposed model to obtain its stability against the small disturbance added into the homogeneous flowunder an open boundary condition. The linear stability analysis shows that the improvement in the stability of traffic flow isobtained by taking into account the power cooperation of the immediately preceding vehicle. The phase diagrams are drawnto illustrate the dependency of the traffic flow stability on the intensity of the preceding power cooperation, the reactiontime, the desired time gap, and themaximum acceleration.We compare the results of the linear analysis with the numericalsimulations for long-wave and short-wave stability.

This paper is organized as follows. In Section 2, the extended intelligent driver model is presented to incorporate thepower cooperation of the nearest preceding vehicle. The linear stability analysis of the proposed model is done with thereaction time of drivers in Section 3. In Section 4, the numerical simulations are carried out to the results of the theoreticalanalysis. Section 5 is devoted to the conclusion.

2. Extended intelligent driver model

The greatest strength of car-followingmodels is that one can easily explore the analytical structure of themodels. As oneof car following models, the intelligent driver model [25–27] can describe the behavior of individual vehicles and drivers bydifferential equations, in which the acceleration of the nth vehicle at time t is determined by the current velocity vn(t), theheadway sn(t), and the velocity difference (approaching rate) 1vn(t) to the leading vehicle:

an =dvn(t)dt

= fn(sn(t), vn(t), 1vn(t)) = α

vn(t)v0

s (t)∗

sn (t)

where α is the maximum acceleration of vehicle, v0 is a desired velocity in free flow, and s(t)∗ is the desired safe headway,which is formulated as the following form [27]:

s(t)∗ = s0 + Tvn(t) +vn(t)1vn(t)

αβ(2)

where s0 is the minimum space gap for completely stopped traffic, Tvn(t) is the velocity-dependent distance, where T isthe constant desired time gap, and β is the desired deceleration. From Eq. (2), one can observe that the last term will workfor a case in which there is velocity fluctuation of vehicle nwith 1vn = 0. Note that, the original IDMmodel is divided intotwo parts: one is the free-road acceleration strategy an = α[1− (vn(t)/v0)4], which only relates with vn(t), the other is thedeceleration strategy an = −α(s∗/s)2 which relates with vn(t), s and 1vn(t).

In this article, we consider the case in which the power output of the immediately preceding vehicle will be an adjustingterm of the power output of the considered vehicle. The considered vehicle pays attention to not only the headway but alsothe power output of the immediately preceding one. If the power output of the preceding vehicle is great, the consideredvehicle assumes that the forward vehicle accelerates, thus it increases the desired velocity even though its headway is short.On the other hand, if the acceleration of the preceding vehicle is negative, the considered vehicle decreases the desire velocityeven if its headway is long.

Now, we try to describe this assumption by the mathematical expression. The power output Fn of the vehicle n is givenby Fn = man, wherem is the mass of each vehicle. Then, the power output Fn of the vehicle based on the power cooperationin front is given by:

Fn = Fn + λFn−1 (3)

528 Z. Li et al. / Physica A 419 (2015) 526–536

Fig. 1. Schematic illustration of the preceding power cooperation.

where the parameter λ is weight of the preceding power cooperation and is assumed to be independent of n. The parameterλ should satisfy 0 ≤ λ < 1 because the dominant part of the power output should be an Fn dependent term.

Then, according to the above discussion and the basic Eq. (1), the extended intelligent driver model with time delay isproposed by the form

dvn(t + td)dt

= fn(sn(t), vn(t), 1vn(t)) + λfn−1(sn−1(t), vn−1(t), 1vn−1(t)) (4)

where td is the physical delay that contains the reaction time andmechanical delay. The advantage of the above form is to beable to explore the effects of the additional term by varying the weight λ. In spite of introducing λ and the preceding powercooperation, the above model provides exactly the same homogeneous flow solution as that of the original one without λdependence, which does not affect it to seek the stabilizing of traffic flow against small disturbance.

The presented new model can be easily carried out by the vigorous development of the connected vehicle technology,which can ensure the real-time exchange of the running information between vehicles by the inter-vehicle communicationsystem. Fig. 1 is a good example of the idea for power cooperation of the nearest preceding vehicle. Driverless vehiclesthat use lasers, in-vehicle communications and radars to sense their surroundings will be able to quickly determine theirpower output and thread their way through traffic. The most critical problem from this assumption is whether it will proveadvantageous for power cooperation to solve some tricky issues in the urban traffic system.

In this paper, we integrate Eq. (4) by the Euler scheme and update the velocity and the position of the vehicle n accord-ing to:

vn(t + 1t) = vn(t) + vn(t) × 1t (5)

xn(t + 1t) = vn(t) + vn(t) × 1t +12vn(t) × (1t)2 (6)

where vn(t+1t) and xn(t+1t) indicate the new velocity and the position of the considered vehicle, vn(t) and vn(t) indicatethe velocity and acceleration of the considered vehicle and 1t is the update time and is selected as 1t = 0.01.

3. Linear stability analysis

In this section, we apply the linear stability method for the extended intelligent driver model described by Eq. (4).It is obvious that the proposed model has exactly the same homogeneous flow solution

xn (t) = (N − n) s + vt, n = 1, 2, 3, . . . ,N (7)

where s is the average headway of adjacent vehicles in homogeneous flow. v is the velocity of vehicles in homogeneous flowand xn (t) is the location of vehicle n at time t . All of these variables have a definite value in the initial state of traffic flow, inwhich all vehicles run with the same headway and the same velocity.

Supposed that yn (t) be the small perturbation from the steady state solution of the vehicle n at time t with the linearFourier-mode expanding:

yn (t) = ceiakn+zt= xn(t) − xn(t) yn → 0 ak =

where c is a constant, and αk = 2πk/N (k = 0, 1, . . . ,N − 1).We conduct the second derivative of both sides of Eq. (8) and we can obtain:

yn(t + td) = xn(t + td) − (xn(t + td))′′ = xn(t + td) =dvn(t + td)

dt. (9)

By substituting Eq. (4) into Eq. (9), one can obtain

yn(t + td) = fn(sn(t), vn(t), 1vn(t)) + λfn−1(sn−1(t), vn−1(t), 1vn−1(t)). (10)

By linearizing the resulting equation, one obtains the following equation

yn(t + td) = f sn (yn−1(t) − yn(t)) + f vn yn(t) + f 1v

n (yn(t) − yn−1(t))

+ λ[f sn−1(yn−2(t) − yn−1(t)) + f vn−1yn−1(t) + f 1v

n−1(yn−1(t) − yn−2(t))] (11)

where f sn =∂ fn∂sn

≥ 0, f vn =

∂ fn∂vn

≤ 0 and f 1vn =

∂ fn∂△vn

≤ 0.

We rewrite Eq. (11) to obtain the difference equation:

yn(t + 2td) − yn(t + td) = td[f sn (yn−1(t) − yn(t)) + λf sn−1(yn−2(t) − yn−1(t))]+ [f v

n (yn(t + td) − yn(t)) + λf vn−1(yn−2(t + td) − yn−1(t))]

+ [f 1vn (yn(t + td) − yn(t) − yn−1(t + td) + yn−1(t))

+ λf vn−1(yn−1(t + td) − yn−1(t) − yn−2(t + td) + yn−2(t))]. (12)

By substituting yn(t) = ceiakn+zt and yn(t) = zceiakn+zt into Eq. (12) and simplifying the resulting equation, one canobtain

(etdz − 1)[etdz − f 1vn (1 − e−2iak) − λf 1v

n−1(e−iak − e−2iak) − f v

n − λf vn−1e

−iak ]

= td[f sn (e−iak − 1) + λf sn−1(e

−2iak − e−iak)]. (13)

Expanding z = z1(iak)+ z2(iak)2 + · · · , etdz = 1+ tdz +t2d z

2 + · · ·, and inserting it into Eq. (13), we obtain the first- andsecond-order terms of coefficients in the expression of z respectively, which are given by

z1 =f sn + λf sn−1

f vn + λf v

n−1(14)

td2 (f v

n + λf vn−1)

z21 −

3λ2 f sn−1

− z1(λf v

n−1 + f 1vn + λf 1v

n−1)

f vn + λf v

n−1. (15)

The uniformly steady-state flow is unstable if z2 < 0 and stable if z2 > 0. Consequently, the stability condition is givenas follows:

f vn + λf v

n−1−

(f sn + λf sn−1)(λfvn−1 + f 1v

n + λf 1vn−1) +

3λ2 f sn−1

n + λf vn−1)

(f sn + λf sn−1)2

. (16)

For the homogeneous flow, f sn = f sn−1, fvn = f v

n−1 and f 1vn = f 1v

n−1 should be satisfied. Thus we make the following simpli-fication for Eq. (16)

f vn (1 + λ)

−f 1vn + f v

f sn (1 + λ)

. (17)

For the specific condition λ = 0, that is no power cooperation and returns the original IDM. We can get it as follows:

f vn f

12(f v

n )2 +12f sn f

vn td − f sn > 0. (18)

What is more, for homogeneous human driven traffic flow without time delays, it is straightforward to show thatEq. (18) is equivalent to:

f vn f

12(f v

n )2 − f sn > 0. (19)

To follow the same solving in Ref. [28], we define:

(ves )

ds= −

f snf vn

. (20)

Simplifying Eq. (19) by using Eq. (20), we can obtain:

(ves )

′ < −12f vn − f 1v

n . (21)

This stability condition is consistent with that in Ref. [28], which is a classic conclusion about car-following model.

For a more intuitive performance about stability condition, we introduce the neutral stability curves in this paper. Inorder to paint stability curves, we must know the specific value about some parameters. According to the stable state of theextended intelligent driver model in homogeneous traffic flow, all vehicles run with the same velocity, so the accelerationand velocity difference are zero. Furthermore, we can get the homogeneous equation as follows:

vn(t)v0

s(t)∗

= 0 (22)

s(t)∗ = s0 + Tvn(t). (23)

From these two equations, we can get some value of determining variables, such as vn(t) and s(t)∗ if we know the head-way sn(t). What is more, if we want to paint stability curves by Eq. (17), we must know the specific form of f sn , f

vn and f 1v

n .So we list them as follows:

f sn =2αn

s0 + Tvn

f vn = −2αn

T (s0 + Tnvn)

f 1vn =

s0 + Tvn

sn. (26)

By substituting Eqs. (24)–(26) into Eq. (17), the stable condition can be further obtained according to the following form:

−2αn

T (s0+Tvn)(sn)2

(1 + λ)

s0+Tvnsn

+ −2αn

T (s0+Tvn)(sn)2

2αnsn

s0+Tvn

2(1 + λ)

. (27)

The phase transition curves can be obtained from the neutral stability criterion (27). Figs. 2 and 3 show the neutralstability curves in the space (v, T ) for different parameters λ with delay td = 0 s and td = 1 s respectively, where themaximum acceleration α = 0.8 m/s2, the desired deceleration β = 1.8 m/s2, free desired velocity v0

= 20 m/s, and theminimum space gap s0 = 1.5 m. The traffic flow is stable with uniform velocity steady state above the neutral stabilityline; conversely, the traffic is unstable and the traffic congestion emerges. From Figs. 2 and 3, one can observe that theneutral stability line lowers with the increase of the weight λ. This means that by taking into account the power cooperationof the immediately preceding vehicle, the traffic flow will be more stable. It is concluded that the incorporation of thepreceding power cooperation stabilizes the traffic flow and suppresses the traffic jams. The effect of the stability of trafficflow by considering the forward power cooperation can be further illustrated in the space (α, T ) in Fig. 4, where the averagevelocity v = 4 m/s, the desired deceleration β = 1.8 m/s2, free desired velocity v0

= 20 m/s, and the minimum spacegap s0 = 1.5 m. From this figure, one can see that by increasing the maximum acceleration α, the neutral stability curvechanges down,whichmeans that traffic flowwill becomemore andmore stable if themaximumacceleration of each vehicleincreases, i.e., the traffic stability can be improved by raising themaximum acceleration. The improvement in traffic stabilityby increasing the weight λ also can be found here, while the delay td enlarges the region of instability.

4. Simulation

To convince the analysis of our theoretical results, we will now solve the extended IDM model (4) numerically underopen boundary condition. Here, wewill perform the simulations for the case in which the leading vehicle of a traffic platoonsuddenly decelerates while the following vehicles respond to this braking action. The traffic disturbance induced by theleading vehicle is created for the purpose of analyzing the stability of the proposed model. At the beginning, a platoon of100 vehicles runs on a single lane without overtaking and inflow, with the same initial headway s = 22.2 m and the sameinitial velocity v = 10 m/s. The vehicles are numbered from the front of the platoon, with the leading vehicle being vehicle1 and the last vehicle being vehicle 100. The initial steady state will be broken when the leading vehicle brakes urgentlyfrom 10 to 4 m/s during the time period 10 s ≤ t ≤ 13 s, followed by the following vehicle moving according to Eq. (4). Thesimulations are carried out by changing the weight λ and desired time gap T . Eq. (4) is calculated numerically through thefourth-order Runge–Kutta method with the time interval 1t = 0.01 s.

Firstly, we examine the effects of the nearest preceding power cooperation on the long-wave stability of traffic flow.Figs. 5–8 show the time evolutions of the acceleration and velocity profiles of 1st, 25th, 50th, and 100th vehicles for the

Fig. 2. Phase diagram in the velocity-desired time gap space for different weights λ with delay td = 0, where the maximum acceleration α = 0.8 m/s2 ,the desired deceleration β = 1.8 m/s2 , free desired velocity v0

= 20 m/s, and the minimum space gap s0 = 1.5 m.

Fig. 3. Phase diagram in the velocity-desired time gap space for different weights λ with delay td = 1 s, where the maximum acceleration α = 0.8 m/s2 ,the desired deceleration β = 1.8 m/s2 , free desired velocity v0

Fig. 4. Phase diagram in the maximum acceleration-desired time gap space for different weights λ and delay td , where the average velocity v = 4 m/s,the desired deceleration β = 1.8 m/s2 , free desired velocity v0

parameter λ = 0, 0.1, 0.2, 0.3, respectively, with the maximum acceleration α = 0.8 m/s2, the desired decelerationβ = 1.8 m/s2, free desired velocity v0

= 20 m/s, the desired time gap T = 1 s, the delay td = 0 s, and the minimumspace gap s0 = 1.5 m. Figs. 5–7 display that the traffic systems are long-wave unstable since the stability condition (27) isnot satisfied for given parameters. For the unstable traffic, the disturbance increases with time as shown in Figs. 5–7. With

Fig. 5. Time evolutions of the acceleration (a) and velocity (b) profiles of 1st, 25th, 50th, and 100th vehicle for the parameter λ = 0, with the maximumacceleration α = 0.8 m/s2 , the desired deceleration β = 1.8 m/s2 , the free desired velocity v0

= 20 m/s, the desired time gap T = 1 s, the delay td = 0 sand the minimum space gap s0 = 1.5 m.

the increase of the weight λ, the amplitude of the traffic disturbance reaching the 100th vehicle decreases. That is to say, byincreasing the power cooperation of the immediately preceding vehicle, the improvement in the long-wave stability of theflow is obtained. The numerical result agrees with the stability condition (27).

Next, let us examine the effect of the power cooperation of the preceding vehicle on the short-wave stability of trafficflow. Figs. 9–11 show the time evolutions of the acceleration and velocity profiles of 1st, 25th, 50th, and 100th vehiclesfor the parameter λ = 0, 0.1, 0.2, respectively, with the maximum acceleration α = 1 m/s2, the desired decelerationβ = 1.8 m/s2, free desired velocity v0

= 20 m/s, the desired time gap T = 2.5 s, the delay td = 1 s, and the minimumspace gap s0 = 1.5 m. We also find that the amplitude of the traffic disturbance reaching the 100th vehicle decreases withthe increase of the parameter λ, which indicates that the theoretical results of linear stability analysis are well consistentwith the simulation of short-wave stability.

5. Summary

In this paper, we have proposed an extended intelligent driver model, which considered the power cooperation of theimmediately preceding vehicle. We have applied the linear stability analysis to obtain the stability condition against asmall disturbance for the extended newmodel under the open boundary condition. It is found that the traffic flow stabilitycan be improved by incorporating the nearest preceding power cooperation, i.e., the introduction of the preceding powercooperation can stabilize the traffic flow and suppress the traffic jams. The direct simulations under the open boundarycondition have been carried out to verify the results of theoretical analysis.We find that the results of linear stability analysisare in good agreement with the simulations for long-wave and short-wave stabilities.

Acknowledgments

Thiswork is supported by the Natural Science Foundation of China under Grant No. 61202384, the Fundamental ResearchFunds for the Central Universities under Grant No. 0800219198, the Natural Science Foundation of Shanghai under Grant No.12ZR1433900, the National High Technology Research and Development Program of China under Grant No. 2012AA112801,and the Scientific Research and Development Program of China Railway Corporation under Grant No. 2013X016-B.

Fig. 6. Time evolutions of the acceleration (a) and velocity (b) profiles of 1st, 25th, 50th, and 100th vehicle for the parameter λ = 0.1, with the maximumacceleration α = 0.8 m/s2 , the desired deceleration β = 1.8 m/s2 , the free desired velocity v0

Fig. 9. Time evolutions of the acceleration (a) and velocity (b) profiles of 1st, 25th, 50th, and 100th vehicle for the parameter λ = 0, with the maximumacceleration α = 1 m/s2 , the desired deceleration β = 1.8 m/s2 , the free desired velocity v0

= 20 m/s, the desired time gap T = 2.5 s, the delay td = 1 sand the minimum space gap s0 = 1.5 m.

Fig. 10. Time evolutions of the acceleration (a) and velocity (b) profiles of 1st, 25th, 50th, and 100th vehicle for the parameter λ = 0.1, with themaximumacceleration α = 1 m/s2 , the desired deceleration β = 1.8 m/s2 , the free desired velocity v0

Fig. 11. Time evolutions of the acceleration (a) and velocity (b) profiles of 1st, 25th, 50th, and 100th vehicle for the parameter λ = 0.2, with themaximumacceleration α = 1 m/s2 , the desired deceleration β = 1.8 m/s2 , the free desired velocity v0

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