RESEARCH OF CONGESTIONS IN URBAN TRANSPORT NETWORK USING CELLULAR AUTOMATON MODEL

. The growing demand for transport and communication services leads to more and more important, traffic related and closely associated problems, especially in the city centre, such as traffic congestion, air pollution, noise and some others. When modelling traffic flows in the Kaunas city centre, the simulation models of crossroads were created based on the principle of cellular automaton model, taking into consideration such relevant traffic indicators as the average speed of traffic flow in different streets, traffic intensity, congestions and distributed flows. Operation of cellular automaton model is associated with the improved approach of further vehicle model. Modelling of microscopic traffic flows is based on the brake light–cellular automaton (BL–CA) model. Random functions were made discreet and autocorrelation values of these functions were calculated in this work. Fundamental macroscopic traffic characteristics were obtained. Numerical dependences of the average traffic speed and traffic flow, traffic density and traffic flow, the average traffic speed and density were determined, with a help of which the demand parameters of transport network were evaluated in the Kaunas city centre in certain time intervals.


Introduction
Today's metropolitan life style requires both collective and individual mobility. For this reason, tra c ows are very important to the national economy. e growing demand for transport network leads to new tra c problems, such as road safety, tra c congestions and air pollution. ese problems are closely connected, consequently, there is a constant tra c ow analysis to reduce indicators and research of new models to nd the best transport system model (Çalışkanelli et al. 2009;Kapski et al. 2008;Nagatani 2005Nagatani , 2006Verhoef, Rouwendal 2004;Wen 2008;Žiliūtė et al. 2010).
Intensity and discontinuity of tra c ow is particularly felt in larger cities, especially in their centres. Various regulations of the European Union, the Republic of Lithuania and the initiated projects emphasize the necessity of information collection related to tra c ows, and the installation importance of mathematical models of tra c ows, their use, integration, evaluation and prediction (Jauneikaitė, Carreno 2009;Jović, Ðorić 2010;Liu 2007;Nagel, Nelson 2005).
In urban centres tra c density is high enough, with such characteristics as low speeds, frequent acceleration and deceleration stop. Over the past decade tra c ow analysis based on cellular automata has increased (Barlović 2003;Lárraga, Alvarez-Icazza 2010). ese models can help to simulate large transport network systems accurately enough. Using the cellular automaton (CA) model it is possible to describe the complex, global and coherent systems and nonlinear phenomena using simple rules. CA models are discrete, which, by their nature, are strictly determined and obeying various laws, but their behaviour can be so complex that it is sometimes di cult to distinguish it from stochastic processes, and the same process may seem chaotic and unpredictable (Boumediene et al. 2009;Gowri, Sivanandan 2008;Lárraga et al. 2005).
CA models are superior to the other tra c ow models for their simplicity because they are single-di-mensional, hardware implementations; they can restore tra c evolution in time of road tra c systems with characterization of tra c ow features; represent the discrete dynamic system, consisting of factors such as discrete grid, cell's state in the grid, cell's neighbourhood, movement in the grid.

Modelling of Tra c Flows in Transport Network
For simulation of tra c ows in transport network the transport system supply and demand is necessary for sources and aggregations of transport network. e model consists of some main elements: zones, nods, connectors, manoeuvres, connecting elements, stops, public transport routes. Nodes indicate the allowed manoeuvre directions and priorities (main, secondary road), street connections are described by the length, number of lanes, speed restrictions on tra c, the tra c throughput (Dell' Acqua, Russo 2010; Dragčević et al. 2008). Modelling has been performed in several stages (Fig. 1). e result of each stage modelling serves as initial data to features modelling. e result of demand modelling is the initial data in the network model, and network modelling results may be initial data of the demand model. OD matrixes in the demand model are evaluated in a way to restore trip times and tra c ows in joints.
Based on the obtained results (trip duration, trafc intensity, tra c ow speed) a state of tra c ows in the concrete parts of transport network system was evaluated. Tra c modelling is based on a number of rules, which are included both into the network and the demand models. With data about transport network system the microscopic tra c modelling based on BL model was performed, and in the result, characteristics of movement of the individual vehicles during the time step were obtained. Later, the statistical analysis of data was carried out and the mathematical techniques for the evaluation of modelling results were applied. From the transport network con guration the macroscopic characteristics of urban tra c were obtained.

Tra c Flow Measurement
e road network of Kaunas region was determined by geographic and natural situations. Kaunas is one of the largest cities in the centre of the country -the main routes from Vilnius, Alytus, Marijampolė, Šakiai, Jurbarkas, Raseiniai, Šiauliai, Panevežys and Ukmergė directions meet there. Tra c ows are mostly formed at the exits and entrances to the city center, where the streets of high-speed meet, which are characterized by high trafc volume -Savanorių Ave, Jonavos St. and Karaliaus Mindaugo Ave, Veiverių St. Vehicle ows moving along them are distributed to several smaller streets of smaller capacity in the Old Town (Šv. Gertrūdos St., Gimnazijos St., Birštono St., Šauklių St.) forming congestions in them (especially in peak hours). e street network of the Old Town is uneven, width of streets varies from 6 to12 meters (even 15 meters wide).
In order to assess the actual load of the existing tra c lanes of street segments and crossroads, the tra c-ow measurements at the crossroad of the main streets of Kaunas city centre were carried out (Fig. 2).
e Old Town is situated in the river valley, builtup by perimeter where tra c ow cause signi cant noise and pollution. e corridor of Šauklių -Šv. Gertrūdos -Gimnazijos Streets is crowded most of the day; it drastically divides the Old Town and causes signi cantly adverse impact on the spatial structure, buildings and people.
Based on data of the modern investment project of management system of integrated tra c, performed several years ago, the max ow is at Šv. Gertrūdos -Gimnazijos St. crossing and makes up 3904 vph. Currently, the max tra c ow in the Old Town is represented by the street sections between Birštono and Gimnazijos St. ( ve lanes) and Šv. Gertrūdos and Šauklių St. (four lanes). Here, the routes from all parts of the city meet -Savanorių Ave, Karaliaus Mindaugo Ave, Jonavos St. and also Vilijampolė bridge connecting the routes of Vilijampolė district. e other smaller streets of the Old Town are completely unsuitable for motor tra c. e attempts to introduce one-way tra c were not successful and did not improve situation.
In 2010 researchers of the Institute of Transport Problems of Kaunas University of Technology carried out the measurements of tra c ows. Vehicles were registered on April 13, 14 and 15 during the morning rush hours from 7:00 to 9:00, during the day time or not rush hours from 11:30 to 13:30 and during evening peak hours from 16:00 to 18:00 with 15-minutes time intervals. Vehicles were separated into di erent classes and di erent tra c directions (BX1, BX2, BY1, BY2, BZ1,  2): motorcycles, mopeds, bicycles, cars, taxi cars, special cars, shuttle vans and other minibuses, light trucks, special light trucks or vans, trucks, multiaxial trucks, buses, city bus service, shuttles tri-axial.

Simulation of Transport Network
CA is a dynamic abstract system, time step is 'T = 1s = (t + 1)t and space is large-grained (Fig. 3). Time axis is pointed downward; space axis extends to the right. CA con guration is given for t and t + 1 time intervals, during which two vehicles n and n + 1 move along the grid. e road is discretized into cells with the length 'X = 1.5 m, vehicle acceleration is equal to 1.5 m/s 2 , acceleration from 0 up to 100 km/h takes 19 s. Vehicle may take a few cells in turn, every cell is empty or it contains one vehicle from N in the transport system. BL-CA model consists of rules (Knospe et al. 2001) used to simulate a complex physical process, taking into account initial conditions of the system. Every n th machine is de ned by two parameters: position x n (t) and speed v n (t) at time moment t. Every next time step t + 1 speed of n th vehicle depends on distance d n,n+1 (t) till next n+1 vehicle moving in front of it, which is de ned by the amount of empty cells. Assessment of the distance d n,n+1 (t) allows to determine whether nth vehicle will stop and activate brake lights and to predict how far n + 1th vehicle will move during the next time step t + 1. e actual distance between n and n +1 vehicle is then de ned by the Eq (1): where: d n,n+1 (t) -distance between the front bumper of n th vehicle and the rear bumper of n+1 vehicle; d safeconstant safe distance. Min movement speed of n+1 vehicle moving in front of: Brake lights b n (t) ^` (0 -turned o , 1-turned on) are the components of predicted movement and allows to respond to the obstacles in front of.
Probability that the n-th vehicle brake, due to n+1st vehicle lights turned on, is determined by the equation: ° where: p b = 0.96 -probability of braking, controlling a back-spreading wave of the brake lights turned on and restoring a synchronized tra c movement; p 0 = 0.5probability that the vehicle will start moving slowly, p d = 0.1 -probability of deceleration, creating possibility for a sudden formation of tra c congestion;  In parallel the BL-CA rules of the same model were applied to all cells and vehicle movement from one cell to another were carried out simultaneously over a single time interval.
Movement of n th vehicle in the system over a time step t +1: ere are two changes of tra c lane: mandatory and discretionary. In the rst case the vehicle is forced to change the lane, for example, when it needs to leave the road or, according to the rules, the vehicle has to follow the lane at the right shoulder.
In the second case the vehicle changes the lane at its own discretion. Vehicles h n n t t or v n (t) > d n,n+1 (t)) and d k,n (t) > v k (t), then j n (t) = right; changing of lane j n (t) = right, n th vehicle changes lane to the right. Judgment on a suitability of the lane changing model is based on certain macroscopic observations. eir example is the lane change frequency, taking into account di erent densities, the ows of each and all trafc lanes, critical density at which tra c in each lane collapses (Angel et al. 2005).
Measurements of the grid of CA models were based on two possible initial conditions: homogeneous initial conditions (evaluated according to studies carried out in the central part of Kaunas city), or compact supercongestion.
In the rst case, all the vehicles were uniformly distributed in the grid and this means uniform intervals of movement forward.
In the second case, all the vehicles were concentrated one a er another with zero intervals. Going from one global density to another, by using equivalent method, the vehicles were adiabatically added (or eliminated) to the already homogeneous or blocked state. Initial conditions corresponding to the rst method were always set for experiments.

Mathematical models
As the result of the latest research the reference (Jablonskytė 2010) widely discusses general vehicle trafc models, analysis of their components, possibilities and conditions of their integration and di erentiation, application possibilities, requiring no de nition of microscopic characteristics of the tra c ow during the trip, under acceleration and deceleration conditions. Emphasis is placed on the need of application of universal criteria and their identi cation.
Suppose we have a random function X(t) (Fig. 4). Consider its two sections -the random variables -X(t) and X(tc) obtained by capturing the di erent moments of time t and tc.
When the value of t and tc is close, the variables X(t) and X(tc) are associated with close dependence: if variable X(t) acquires any value, it is highly probable that the variable X(tc) will acquire value close to that one. It follows that with the increasing interval between sections t and tc, the dependence between the variables X(t) and X(tc) may be lost.
Degree of dependence of variables X(t) and X(tc) can be characterized by their correlation moment, which is the function of two arguments t and tc. is function is called the correlation function.
us, the correlation function of the random function X(t) is called not random function of two arguments K x (t, tc), which for each pair of time values t and tc is equal to the correlation moment of obtained sections: where: X t X t m t X t X t m t . Suppose, that tc = t, then: where: D x (t) -dispersion of obtained section. us, there is no need to consider dispersion as the random feature. For investigating the essential character points of the random function it is su cient to consider only its average and correlation function.
Instead of the correlation function K x (t, tc) we may use the rationed correlation function: which determines correlation coe cient of variables X(t) and Xc(t). When tc = t is the rationed correlation function and is equal to 1.
When X = (x 1 , x 2 ,…,x n )n-dimensional random vector with the vector of mean values M = (m 1 , m 2 , …, m n ), then the vector's correlation function is (Jarner, Tweedie 2001): e random vector X is normal, if its n-dimensional probable density is described by formula: where: K -determinant of the correlation matrix K of vector X. Vector X is the vector of values of random process X(t) in points t 1 , t 2 ,…,t n , thus 1 2 ( ), ( ), , ( ) . n X x t x t x t ! e process X(t) is considered to be stationary and normal process and is completely characterized in correlation theory, or the multidimensional normal distribution function is de ned, if moments of its rst two rows are known.

Results and Discussions
e results of investigation showed that the crossing Gimnazijos St. -Šv. Gertrūdos St. (Fig. 5) is mostly crowded during the morning rush hour from 7:45 to 8:00 and during evening peak hours from 17:00 to 17:15. e prevailing group of the mixed tra c ow was cars, making 70÷80%. e other group of vehicles of lower manoeuvring speed and at rst glance seeming insigni cant, in uences the velocity of ow and the capacity of tra c lane. e max ow was recorded in BX1 and BY1 directions in all three periods investigated. During the morning rush hour tra c intensi cation was in BX1 direction, the highest number of cars from 7:30 to 8:15 was 180 vehicles per hour. A number of buses, trolley buses, minibuses and trucks during that period was quite stable. e morning rush hour starts at 7:30 and lasts till 8:15.
During the day and evening peak there was no pulsation in this direction -a number of cars, buses, trolley buses, trucks and minivans over 15-minutes intervals was almost invariable. In the morning a number of vehicles travelling straight (from Šauklių St. towards Savanorių Ave) was signi cantly higher than of those turning to the right (from Šv. Gertrūdos St. to Gimnazijos St.). Over two hours 1942 vehicles passed in these directions, i.e. 1299 vehicles travelled straight and 643 vehicles turned right. About 80% of all vehicles going straight were passenger cars, the cars turning right made up 90%.
Assessing the results presented, it could be concluded that during the day a number of vehicles going straight is nearly twice higher than those turning le . During these two hours 1240 vehicles were going straight and 679 vehicles were turning le . In the evening peak, like in the morning and a ernoon peak hours, 80% of vehicles travelling straight and 90% of those turning le were passenger cars. During evening peak hour a number of buses and trolley buses, going from Šauklių St. towards Savanorių Ave, have increased signi cantly in comparison with the day period. During this period, 80 buses and trolley buses were recorded. e most acute ow direction in sense of capacity at the Gimnazijos St. -Šv. Gertrūdos St. crossing is Simulation of microscopic tra c ows is based on BL-CA model because of the possibility to restore road tra c evolution in time and space. e model to evaluate the distances between moving vehicles, the average vehicle speed in separate road segments, lengths of turns due to stop at prohibited tra c lights, etc. was set up.
Assessing changes of the vehicle speed in road segment of one lane, it was found out that average speeds are almost identical under the same tra c conditions and do not depend on random realization. But modelling sections of transport systems with few or more lanes provides a completely di erent behaviour, and sequential realizations of several simulations have also given completely di erent results.
Trying to solve the problem possibilities of vehicles the turnings le and right were taken into account. When a certain number of vehicles are turning le or right, behaviour of BL-CA model is obtained even. If the restrictions, mentioned earlier, are included into the model, behaviour of the model is similar to di erent tra c ows in case of one lane and also in cases of few or more lanes.
Evaluating lane-changing rules with the help of the CA model, several conditions describing this manoeuvre were determined. It was found that the vehicle is typically changing lanes during successive time intervals. In order to avoid that, a coincidence of lane change was introduced, thus, unrealistic e ects that appear in transport system were destroyed. e analysis of individual vehicles movement in the transport system gave information about the drivers' behaviour. Vehicles autocorrelation and approximated autocorrelation functions are presented in Fig. 6. Autocorrelation was de ned taking into consideration the average movement speed of individual vehicles over time interval which is equal to one minute, strong correlation among the speed of individual vehicles was then de ned. e value of autocorrelation function D = 1.899 was determined by the leastsquares method, where con dence interval is (0.1214; 0.2813) and con dence level is 0.99. Empirical values of autocorrelation functions were approximated by the obtained phenomenon: Correlation disappears when traffic density in transport system is very small, i.e. 0÷10%. Strong correlations were obtained at small time intervals (Fig. 4), this explains the intense movement of vehicles in transport system, when distances between them are minimal or vehicles in the system move in small groups.
Macroscopic characteristics: densities, ows and average speeds in CA grid L, obtained from local measurements with an arti cial loop detector of unit length. To ensure that no vehicle could 'avoid' the detector be-tween consecutive time intervals, local measurements of tra c ows and densities over certain time interval T mp were done. An average ow speed is derived using the main ratio of tra c ow theory (Kerner 2010;Nagel, Nelson 2005): where: q -tra c, veh.numb./h; k -tra c density, veh. numb./km; v s -average ow speed, km/h. Diagram of the average tra c ow speed and tra c ow in Gimnazijos St. segment, corresponding to the fundamental Kerner diagram, is presented in Fig. 7. Dependence of an average tra c speed and tra c ow in Gimnazijos St. is square: With free movements of vehicles, tra c ow speed increases until a threshold of tra c congestion, then the tra c starts to diminish, while tra c intensity is growing quickly. With the growth of tra c ow, an average ow speed in system cuts up to ~ 3 km/h, which means a great congestion of vehicles in transport system and almost immobile vehicles.

Conclusions
e analysis of tra c ows in urban transport network and formation of illustrated congestions which appear for a variety of reasons such as urban services, inadequate transport infrastructure and inadequate street intersections; lack of trunk-streets of continuous tra c; irrationally used carriageways of congested streets were presented. e largest tra c ows in Kaunas city appear at the exits and entries to the city centre, where streets of high capacity, characterized by high tra c volume meet -Savanorių Ave, Jonavos St. and Karaliaus Mindaugo Ave, Veiverių St. Vehicle ows moving down them are distributed to several smaller streets of lower capacity of the Old Town (Šv. Gertrūdos, Gimnazijos, Birštono, Šauklių Streets) causing congestions on them (especially in peak hours).
e results of tra c-ow measurements showed that the Gimnazijos St. -Šv. Gertrūdos St. crossing (Fig. 5) is mostly crowded during the morning rush hour from 7:45 to 8:00 and during evening peak hours from 17:00 to 17:15. e prevailing group of the mixed tra c ow is cars, making 70-80%. e most acute tra c ow direction in terms of the capacity is crossing Gimnazijos St. ow, turning to Šv. Gertrūdos St.: 425 vehicles run in this direction at 17:15÷17:30. e programme tool for simulation of microscopic tra c ows and evaluation of their parameters in microscopic and macroscopic scales, highlighting the tra c ow modes and de ning of transition boundaries between the tra c modes has been compiled. Correlated vehicles speeds lead to movement of vehicles in a stable transport system, where the con dence level is 0.99.
With application of the distances between moving vehicles, average vehicle speeds in individual road segments and the microscopic cellular automaton model in nonlinear dynamic system, evaluating row lengths formed at tra c lights prohibited, the relationship between tra c ow and tra c density, tra c ow and average ow rate has been determined.
According to the tra c congestion dynamics the Fundamental Kerner diagram has been created, causing the relevance of the model to simulate the urban transport network system.
With de nition of a separate vehicle dynamic characteristics and technical parameters, test results and the created so ware tool should be assessed as initial data in investigating or predicting congestion formation at nano-level.