Microscopic traffic simulation tools and their use for emission calculations

Transcription

1 Microscopic traffic simulation tools an their use for emission calculations Stephan Rosswog & Peter Wagner, MS, DLR, Köln-Porz an ZAIK, Uniersity of Cologne Nils Eissfelt, ZAIK, Uniersity of Cologne I. Introuction The basic application areas of traffic simulation tools are congeste urban areas. The fact that here typically 10 5 to 10 6 iniiual rier-ehicle units an their mutual interactions hae to be moele an simulate unerlines the necessity for a restriction to the ery basic features of these interaction to accomplish maximum efficiency (hoping that macroscopically releant features are still reprouce). It is this restriction to minimalistic moels that makes the tackling of realistic, often large-scale, problems possible. To this aim a moel chain has to be built up whose ingreients shoul be as simple as possible, but not simpler. The first step of this chain is the knowlege of traffic eman in the system uner consieration. This is a basic input to eery traffic simulation an the latter one can only be as goo as the input knowlege about the trael emans. This knowlege is obtaine either by performing large-scale emographic inquiries or moeling synthetic populations or combinations of both. Once the traffic eman is known it is transforme into a (for our purposes ynamic) trip tables which contains the trip rates for specifie origin-estination pairs, i.e. it contains information on how many people want to trael at a gien time t k from one origin i to a gien estination j. The next question concerns the istribution of the trips to the aailable transportation moes, the so-calle moal split. The following block in the moel chain has to mimick the route choice behaior of the traelling iniiuals. The basic unerlying assumption here is Warrop s first principle, i.e. one assumes that noboy can change unilaterally his or her route an improe the corresponing trael time. Once all the routes (or the set of possible routes, see below) is known, the microscopic (the notion microscopic refers in this context to the treatment of iniiual rier-ehicle units) riing moel enters the game. In this paper we will focus exclusiely on the last two points, i.e. the route choice an the microscopic riing moeling an possible applications to the moeling of emissions. II. Moel ingreients II.1 An efficient algorithm for Dynamic traffic assignment Assume you moe to a new city that is unknown to you an you hae to fin your way to the place where you work. How woul you procee? First you woul start with some kin of a sophisticate guess, i.e. you woul either consult a map to take the way that looks shortest or you woul ask people with a better knowlege of the localities. Howeer, it may be possible that this first guess for the route passes through a region that is prone to congestion like for example the center of the city an that you en up reaing your newspaper while staning in a traffic jam. The next morning you woul take a possibly neighboring, but smarter route to improe on your trael time. This will procee until it seems impossible to further improe your trael time by choosing a ifferent route.

2 The approach to be escribe here (for further etails see [1,2,3]) is base on a ery similar proceure, each iniiual is proie with a set of shortest routes, tries them out an then chooses the ones that take the shortest trael times. Each rier-ehicle unit is escribe by the following ariables: an origin O, a estination D an a eparture time t, a set of routes P leaing from O to D, a probability istribution p on the set of routes, i.e. a function p : P [0,1] with p ( r) = 1, which gies the probability that the rier chooses a route r, r subjectie costs : c : P R + which represent the rier s knowlege of the trael times on each route. It is not possible to enumerate all possible routes in an efficient way, thus the set of routes contains only a small subset of all possible routes. These are chosen in the following way: (i) etermine the k shortest paths that o not use any noe more than once, (ii) if the trael time spent waiting in a queue excees a gien limit new routes may be ae. Once this is one the following steps are iterate until conergence is reache: 1. choose a route r with a probability p (r) 2. simulate the motion of the ehicle-rier units accoring to these routes following a prescribe microscopic traffic moel an upate the memorize trael costs 3. shift the probabilities towars cheaper routes ( learning ) 4. if necessary a a new route to P. Keeping in min that the consiere problems inole seeral 10 5 learning iniiuals an that seeral iterations will be neee to fin the user equilibrium suggests using some low fielity moel for the motion of iniiuals through the network, otherwise most realistic problems are not treatable. Therefore a queuing moel [1] is use for the motion of the ehicles along the links uring this route choice iteration. When a ehicle enters a link its trael time is calculate accoring to its elocity an the length, capacity an the number of lanes of the link. Each time step a certain number of ehicles that has arrie at the en of the link is rawn ranomly an transmitte to the next link (the rest stays in a priority queue), where the restrictions come from the capacity of the link an the number of cars that fit into the next link. This minimalistic riing moel is extremely efficient an its efficiency grows with the aerage link length. For the freeway network of the German Buneslan Norrhein-Westfalen, for example, with an aerage link length of 2.4 km the speeup in comparison with the ery efficient Nagel- Schreckenberg moel (CA; [4]) is almost two orers of magnitue. The question of the conergence of the route choices naturally arises in this context. Due to the stochastic nature of the unerlying algorithm only a stochastic equilibrium (in a sense that the aerage trael time conerges but iniiual trael times still fluctuate) is obtaine. All inestigate network examples showe conergence in this sense. For the network of the German city of Wuppertal (9098 noes, links an trips) conergence was achiee after 35 iterations, each iteration took approximately 2 hours of CPU time on a 250 MHz UltraSPARC. Howeer, the uniqueness of the foun solution cannot be claime, since the necessary conitions will generally not hol for our simulation base cost functions. II.2 Microscopic Driing moel The aboe escribe queuing moel works fine for the rout choice algorithm, howeer, for the calculation of emissions a more elaborate moel has to be use. If the routes of all the riers are known the question which microscopic riing moel to use arises. A ery efficient, minimalistic moel is the Nagel-Schreckenberg moel [4]. It has been use for a large number of traffic relate problems (for an extensie reiew see [5]). Howeer, in this moel

3 (at least in its most basic ersion) accelerations are boune by the time iscretisation alone, i.e. they can be as high as V max / t (where V max is the maximum elocity an t the time step), i.e. they can ierge in the continuous limit. Since emissions are strongly non-linear functions of the accelerations, this moel cannot be applie in a straight forwar way for realistic emission calculations. For recent attempts to use CA-moels for emission calculations see [6]. Krauß [7,8] has eelope a realistic microscopic moel that results by construction in a realistic acceleration behaior. It is erie from the following assumptions: 1. elocities are boune by Vmax 2. accelerations a as well as ecelerations b are finite 3. ehicles rie in a way that collisions are aoie. The safety conition reas: ~ ( ) + τ ( ~ ) + g, where / ~ ~ are the elocities of the following/leaing ehicle, () / ( ~ ) are the corresponing braking istances, τ is the reaction time of the following ehicle an g is the gap between both ehicles. x, ~ x, ~ g Figure 1: Quantities of the Krauß moel After solution for the sae elocity for the following car safe an transformation to a map (iscrete time) the following upate rules are foun es safe = min[ ( t) + a, ~ g τ = + / b( ) + τ ( t + t) = max[0, safe, es max ] εaζ ] where = ( + ~ ) 2, ε a parameter of orer unity an ζ is a ranom ariable uniformly istribute in [0,1]. This means that the riers follow some kin of a stochastically perturbe maximum-butsafe-elocity strategy. The moel is only aroun 30 % slower than the CA moel an thus excellently suite to be use for large-scale problems. If ehicles can accelerate only moerately, i.e. for small a, it takes longer to recoer from being slowe own ranomly. Thus ranom perturbations surie longer in this case which faors structure formation for low alues of the acceleration parameter a. The parameter b basically etermines some kin of interaction length since it goerns the istance to the preceing ehicle. For low alues

4 of b it takes ehicles a long istance to come to a stop, thus the no-collision-conition leas to large gaps between subsequent ehicles for this case. Depening on the choice of the parameters a an b three ifferent moel classes are foun: in class I ehicles always react fast enough to aoi jamming, thus no structure formation is foun here. In class II an III structure formation ( spontaneous jamming ) is encountere. Howeer, the metastable states of high flow that are obsere in reality are only foun in class II, which makes this class with low alues of a an b the most realistic one. a I III II b Figure 2: Different moel classes as a function of acceleration/eceleration parameters III. Application to emission moeling Traffic emission rates are strongly non-linear functions of the current state of motion of the iniiual ehicles, namely they epen enormously on the current acceleration of a ehicle. Thus it is inispensable to perform emission calculations with microscopic moels that show realistic elocity changes. For reasons of illustration we show the elocity as a function of time of a single ehicle for two ientically prepare initial traffic states (close loop, 5000 ehicle, max = 150 km/h), but once the ynamics follows the upate rules of a cellular automaton (CA; braking probability p brake = 0.2) an the other time the ehicles are upate accoring to the space continuous Krauß-moel (SK). In both cases spontaneous structure formation is encountere (the so-calle jam out of nowhere ). In Figure 3 spontaneous jammings are compare for both moels. While for the CA moel an enormous amount of short-lie microjams forms (these go along with immens accelerations an ecelerations) the SKmoel exhibits a much more reasonable riing behaior, which shows a breakown to practically zero elocity on a time scale of aroun 25 secons an then a recoering phase where the maximum elocity is regaine on a time scale of aroun 40 secons. This realistic riing behaior of each ehicle makes this moel an excellent caniate for emission calculations. For this purpose a elocity epenent emission table was create whose entries represent aerages oer typical accelerations. Couple to the before mentione moel blocks emissions can be calculate irectly uring a network simulation. As an example Figure 4 shows the cumulate NO emissions for a simulation of the street network of the city of Cologne. x

5 Figure 3: Comparison of jamming behaior of the cellular automaton (CA) an the Krauß moel (SK) $ $ $ # # $ # $ " # $ " $! # # # # # $ # $ # # % # % # # & $ $! # " N A E I I E # C H C D > N Figure 4: Total NOx emissions for the street network of the city of Cologne

6 IV. Summary We hae presente basic ingreients of a moel chain that allows for an efficient simulation of the ynamics an the resulting emissions of traffic flow in large street networks. Using etaile knowlege of ynamic origin estination relations of the system uner consieration as input we hae presente an efficient algorithm that fins a simulation-base stochastic ynamic user equilibrium. The algorithm mimicks the learning behaior of iniiual rier-ehicle units an proies each unit with a set of possible routes an the corresponing probabilities to choose one of them. For all inestigate systems conergence is achiee within a reasonable amount of computing time. Once each ehicle is proie with these routes it has to be moe through the loae network. The approach escribe here is some kin of a stochastically perturbe optimal strategy of each rier. It is base on three assumptions (i) there is a maximum elocity, (ii) each rier ries in a way that collisions are aoie thereby accounting for his or her (iii) boune acceleration/eceleration capabilities. The transformation of these assumptions into a (ery efficient) algorithm enables the simulation of traffic flow that reprouces obsere macroscopic properties ery well an in contrast to the minimalistic cellular automaton moel- also exhibits a ery reasonable riing behaior of iniiual ehicles. This realistic acceleration/eceleration behaior combine with its extreme numerical efficiency makes this moel an excellent caniate for the calculation of pollutant emissions. References [1] C. Gawron, An iteratie Algorithm to etermine the ynamic User equilibrium in a Traffic simulation Moel; Int. Jrn. Mo. Phys. C9(3): , 1998 [2] C. Gawron, S. Krauß, P. Wagner, Dynamic User Equilibria in Traffic Simulation Moels, in Traffic an Granular flow 1997, p , Springer, Singapore, 1998 [3] C. Gawron, Ph.D.-thesis, Uniersität zu Köln 1998 [4] K. Nagel, M. Schreckenberg, A Cellular Automaton Moel for Traffic Flow, J. Physique I2: 2221, 1992 [5] D. Chowhury, L. Santen, A. Schaschneier, Statistical Physics of Vehicular Traffic an some Relate Systems, submitte to Phys. Rep [6] M.D. Williams, G. Thayer, M.J. Barth, L.L. Smith, The TRANSIMS approach to emission estimations, LA-UR , [7] S. Krauß, Ph.D.-thesis, Deutsches Zentrum für Luft- un Raumfahrt (DLR), Köln-Porz [8] S. Krauß, P. Wagner, C. Gawron, Metastable states in a microscopic moel of traffic flow, Phys. Re. E 55, 5597 (1997)