Available online at www.sciencedirect.com
ScienceDirect Transportation Research Procedia 20 (2017) 676 – 682
12th International Conference "Organization and Traffic Safety Management in large cities", SPbOTSIC-2016, 28-30 September 2016, St. Petersburg, Russia
Features of Calculation of Traffic Light Control Modes in the Conditions of Intensive Road Traffic Alexey Vlasov * Penza State University of Architecture and Construction, 28 Germana Titova Str., Penza, 440028, Russia
Abstract Controlling intensive traffic flow differs radically from free flows control, while management of transport networks can be complicated due to inability to locate excessive density of flows within a single intersection. This paper applies a model of cyclic balance of transportation demand and supply in order to find out an optimum traffic light control mode. The traffic light control mode is determined by challenging the issue of maximization of traffic intensity control subject to restrictions on the volumes of arriving and leaving flows and the queue length at respective sets of links. A feature of the proposed method is a differentiated approach to transportation links. It suggests that excess traffic demand in the control zone is localized on a set of links characterized by presence of a controllable line. © Published by Elsevier B.V. This ©2017 2016The TheAuthors. Authors. Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 12th International Conference "Organization and Traffic Peer-review under responsibility of the organizing committee of the 12th International Conference “Organization and Traffic Safety Safety Management in large cities". Management in large cities” Keywords: coordinated traffic lights; intensive traffic; congestion; intensity balancing; coordination; control area
1. Introduction Controlling intensive traffic flow differs radically from free flows control, while management of transport networks can be complicated due to inability to locate excessive density of flows within a single intersection, where a road block originated [Lagerev et al. (2010), Zhivoglyadov and Bakhtina (2004)]. A negative aspect of connection between separate intersections is that increasing queue at one intersection blocks traffic flows of the previous
* Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address:
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2352-1465 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 12th International Conference “Organization and Traffic Safety Management in large cities” doi:10.1016/j.trpro.2017.01.110
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intersection as viewed in the direction of the flow moving, which, in its turn, can block operation of the adjacent intersections, etc. This process can completely paralyze movement at a separate area of the transport network for a long time. 2. Main text Considering the above, the main task of control systems in the conditions of intensive traffic flows should be prevention of public traffic congestions and management of consequences in case of traffic jamming [Kapitanov and Khilazhev (1985), Pechersky and Khorovich (1979)]. The obvious way of solving the problem is to eliminate the causes of jamming in narrow spaces of the network. Since the control system is not able to increase the capacity of relevant intersections, the only way to reduce the risk of congestion is timely limiting of the number of transport arriving to a risky network section [Orlov et al. (2014), Vlasov (2012), Bielefeldt and Busch (1994), Hunt et al. (1981)]. 2.1. Model of Transport Network Let us consider formation of traffic flows within an area of the transport network at a constant transport demand. The transport network can be presented as a directed graph with intersections as its nodes and mid-points of intersections as its arcs. We introduced the following notations: J is the set of transportation links in the control area; N is the set of controlled intersections. The model of traffic formation within the intersection (Figure 1) describes conversion of the arriving traffic flow
qaarriv into the leaving traffic flow qbleaving
[Lin and Xi (2008)].
2
q2 , 3
q1,2
3 1
q1arriv
q1,3
6
q3leaving
q1, 4
q4,3 4 Fig. 1. Originating of traffic flows at the intersection.
qaarriv arriving at the intersection is divided into m flows according to the following formula: (1) ¦ Ea,bqaarriv ;
Each traffic flow
qaarriv where
arriv Ea,b is the ratio of the arriving flow qa , moving from the link a to the link b,
set of links involved at the intersection,
aJ ;
b is the set of links of the leaving flow,
¦E
a ,b
1.0 ; a is the
b J ;
J is the set of
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transportation links at the control area. Intensity of the traffic flow leaving the intersection by the link b will be defined as the sum of turning flows onto that link:
¦E
qbleaving
arriv a,b a
q
(2)
a
Active traffic light control imposes restrictions on the number of cars passing through the intersection. In this case, intensity of the leaving flow is determined on the basis of duration of traffic light phases in a cycle according to the following formula:
§
¦ min ¨ E
qbleaving
a
where
cа
©
arriv a ,b a
q
,
ca gn,a · ¸ Tn ¹
is intensity of the flow a;
(3)
gn,a is duration of the enabling signal on the link a of the intersection n N
; Tn is duration of a traffic light cycle at the intersection n. Let us consider formation of a network traffic flow by the example of a road section comprising two adjacent intersections (Figure 2). 5
2
q2 , 3
q5,6
3
1
q3leaving
q1arriv
6
q3arriv
q6leaving
q4 , 3
q7,6
4
7 Fig. 2. Formation of traffic flows in the transportation network.
The traffic intensity was defined precisely on external links of the network {1, 2, 4, 5, 7} The intensity of road traffic .on the inside and leaving links {3, 6} depends on the mode of traffic lights. Moreover, if in case of the link 3 all the arriving flows are known for calculation of traffic intensity, the link 6 requires determination of the intensity
q3arriv unknown at the beginning of the calculation.
Assessment of traffic intensity for the whole set of links J cannot be performed under consideration of the impact of all the traffic lights of the stated network. All the values of traffic intensity at all the links can be obtained by consequent identification of turning flows available for calculation. The iterative procedure would be as follows: 1. Defining the set QJ, including links with definite traffic intensity 2. Defining the set MJ, including all the traffic links with indefinite traffic intensity 3. Calculation of arriving flows for the set of links M by the formula (2) 4. Calculation of leaving flows involved in the set of links M with defined arriving links is performed by the formula (3) followed by their transfer into the set Q 5. If the set Q is not empty, refer to Item 3. Otherwise, the procedure is completed. As a result of the procedure we obtain a set Q with defined arriving flows taking into account the control system g.
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2.2. Formulation of Optimization Task Let the condition of the transportation network be estimated with the functional Φ
u t , x t . The task of
control optimization for the traffic area would be as follows:
Φ u t , x t o min max
(4)
umax t u t t umin
(5)
xmax t x t t xmin where
u t
(6)
is the control vector;
controlling impacts;
xmax
and
x t
is the status vector of the network;
umax
and
umin are restrictions of
xmin are restrictions of the network status.
In general, the cause of traffic jamming is residual queues at the intersection j as a result of excessive number of cars coming to
qleaving j
compared to the number of cars
qarriv that j
have left the queue. Let us define the set of
“secured” transport links J J , where it is necessary to ensure free traffic without congestions. It is possible to prevent congestions on the intersection j pr J pr by valuation of the number of cars arriving onto it. The purpose of such control would be passing of the most number of cars through the links. The tasks (4–6) would take the following form: pr
¦ q u t , q arriv j
leaving j
t o max
(7)
J
under the following constraints:
qleaving t qarriv t d 0 ; j pr j pr
(8)
umax t u t t umin
(9)
mj t lvech d Lj
where
¦q u t , q J pr
arriv j pr
leaving j
t is total intensity of cars leaving at the control area,
intensity of traffic flows arriving into the transportation node; transportation node; the queue;
L j is the length of the link j; mj t
(10)
jJ ; q
leaving j pr
t is the
qarriv t is the intensity of traffic flows leaving the j pr
f u t , qleaving t , qarriv t is the number of cars in j j
lvech is an average length of a vehicle.
The stated formulation of the task of traffic lights control ensures balancing of transportation demand and supply within a separate area of the transport network. Restrictions (8) and (10) guarantee the absence of network blocks in the control area. However, there are such conditions of a transport network when the task (7) has not solutions satisfying the specified constraints. Solution of this problem may include introduction of restrictions on the intensity of traffic flows arriving into the control area. Let us consider such traffic flows in the network comprising two traffic areas A and B (Figure 3) having common transportation links edg+ and edg–. The link edg- is the leaving one for the traffic area B (traffic flows leave the area with the help of this link), while the link edg + is arriving one (traffic flows arrive to the area with the help of it), respectively. Let us introduce the set of arriving transport links I J having unlimited allowance for queue growth in order to obtain a correct solution of the control optimization task (7) in the area B and then introduce edg + into it. B
B
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А
B edgedg+
Fig. 3. Scheme of a transportation network with two control areas.
Excessive transportation demand for the link edg+ would result in the residual queue following equation:
A B Qo t max 0, Qo qedg t t c edg
Qo is
where
the initial residual queue;
transportation link edg +;
Qo t determined by the
A qedg t is
(11) the intensity of traffic flow leaving the area A by the
B cedg t is the road capacity of the link edg + considering control in the area B.
Balancing of transportation demand and supply between the areas A and B would be possible by fulfillment of the
t 0 . The task of traffic control of the area A would become as follows: t o max u t , qleaving ¦qarriv j j
condition Qo
(12)
J
under the following constraints:
qleaving t qarriv t d 0 j pr j pr
(13)
umax t u t t umin mj t lvech d Lj
(14)
A B Qo qedg t d 0 t c edg
(15) (16)
Determination of the task for effective control (13–16) provides a balance of transportation supply and demand both within the control area and at external links, allowing identification of possibility of traffic congestions in the network. 2.3. Network Control Optimization and Simulation The task of effective transport network control defined in the equations (7–10) is the task of nonlinear programming, which was solved with the help of a module of genetic algorithm of OpenOpt pack. Let us consider a solution of the task of effective control of the network, shown in Figure 4. Cars pass the intersections A, B and C by four steps (left-turn steps are marked). The intersection D assumes a fifth pedestrian step.
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C
400 m
300 m
600 veh/h
475 veh/h
550 veh/h
1150 veh/h
400 m
D
2000 veh/h
B
600veh/h
625 veh/h
500 veh/h
575 veh/h
2500 veh/h
A
Fig. 4. Scheme of the network.
A microscopic network model designed in SUMO application was used to evaluate the efficiency of control [Krajzewicz et al. (2012)]. A well-known adaptive control method (actuated traffic control algorithm) was applied as a basic method of control adopted. The basic control method applied at the space interval of C–D revealed a congestion which expanded all over the network in 20–25 minutes, blocking the traffic route A–B–C–D. Using the traffic light control mode, obtained by solution of equations (7–10), there were no congestions occurred on the route A–-B–C–D. At that, delays on the route A-B-C-D reduced by 2.5–4 times, and the average travel time in the network reduced from 234.63 seconds to 211.67 seconds. 3. Conclusions As the above stated calculations showed that the proposed formulation for solution of traffic lights control optimization under consideration of the duration of the control cycle degenerates into the value of the maximum duration of the control cycle. Thus, it is advisable to use it to solve the problem of effective distribution of the cycle duration between the steps. The further development of the calculation method for traffic light control modes would be its fitting for real-time operation. The author is carrying out appropriate investigations of control with a forecasting model (“model predictive control”) to control the traffic intensity. Acknowledgments The work is performed under the support of the Innovations Assistance Fund (contract No. 509GС1/9720). References Bielefeldt, C., Busch, F. (1994). MOTION — a new on-line traffic signal network control system. In proceedings 7th International Conference “Road Traffic Monitoring and Control”, London, pp. 55–59. Hunt, P. B., Robertson, D. I., Bretherton, R. D., Winton, R. I. (1981). SCOOT — A Traffic Responsive Method of Coordinating Signals. TRRL Report 1014. Transport and Road Research Laboratory, Crowthorne, Berkshire, UK. Kapitanov, V. T., Khilazhev, Ye. B. (1985). Management of urban traffic flows [Upravlenie transportnymi potokami v gorodah]. Moscow: Transport, (in Russian). Krajzewicz D., Erdmann J., Behrisch M., Bieker L. (2012). Recent Development and Applications of SUMO — Simulation of Urban Mobility. International Journal On Advances in Systems and Measurements, 5 (3&4): 128–138. Lagerev, R. Yu., Mikhaylov, A. Yu., Lagereva, S. V. (2010). Methods of prevention of transport network blocks [Metodika preduprezhdenija setevyh transportnyh zatorov]. Bulletin of the Scientific Center of Children’s Safety [Vestnik NCBZhD], (5): 82–88 (in Russian).
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Lin, S., Xi, Y. (2008). An efficient model for urban traffic network control. Proceedings of the 17 th World Congress. The International Federation of Automatic Control, pp. 14066–14071, Seoul, Korea. Orlov, N. A., Vlasov, A. A., Chushkina, K. A. (2014). Synchronizing of traffic lights modes under intensive traffic conditions [Sinhronizacija raboty svetofornyh obektov v uslovijah nasyshhennogo dvizhenija]. Modern Problems of Science and Education, No. 2 (in Russian). Pechersky, M. P., Khorovich, V. G. (1979). Automated traffic control systems [Avtomatizirovannye sistemy upravlenija dorozhnym dvizheniem]. Transport, Moscow (in Russian). Vlasov, A. A. (2012). Urban adaptive traffic control systems [Adaptivnye sistemy upravlenija dorozhnym dvizheniem v gorodah]. Publishing House of the Penza State University of Architecture and Construction, Penza (in Russian). Zhivoglyadov, V. G., Bakhtina, O. N. (2004). Theoretical principles of occurrence and prevention of jamming on roads [Teoreticheskie principy vozniknovenija i uprezhdenija zatorovyh sostojanij na avtodorogah]. University News. North-Caucasian Region. Technical Sciences Series, No. 3, pp. 103–105 (in Russian).
