ONLINE OFFSET OPTIMISATION IN URBAN NETWORKS BASED ON CELL TRANSMISSION MODEL Essam Almasri, Bernhard Friedrich Ph.D. Student, Professor Institute of Transport, Road Engineering and Planning University of Hannover, Appelstr. 9A, 30167 Hannover, Germany E-mail:
[email protected],
[email protected]
SUMMARY This study presents a heuristic approach for online offset optimisation in road networks. Traffic flow is modelled using the cell-transmission model (CTM), which is a discrete approximation to the hydrodynamic model of traffic flow. The optimisation employs a Genetic Algorithm (GA) and is applied to a two-way urban road with 3 signalized intersection and to a realistic small network with 6 signalized intersections. This paper presents the theoretical formulation of the traffic model and the optimisation algorithm, shows up figures on performance measures in terms of travel time savings, and addresses computational issues like CPU-time.
INTRODUCTION Adaptive online traffic control in urban road networks requires information on the present and future traffic state. This information needs to be as complete and precise as possible. For this reason advanced control methods involve respective traffic models in order to complement local measurements and to derive criteria on the traffic state that cannot be measured. As criterion to be optimised most models calculate either the queue length (e.g. (5), (6), (8), and (9) ) or the number of vehicles per link (e.g. (4) ). It is therefore the precise determination and short term forecast of the mentioned criteria which is a crucial prerequisite for any adaptive control strategy. For optimisation purposes in online traffic control methods either equilibrium queuing models or discrete models can be employed. Whereas equilibrium models can only be applied for macroscopic adaptation of split and cycle time at single intersections, discrete models also allow for microscopic adaptation and in particular for offset optimisation. However, the employment of models, discrete in time and space, is constrained by the run-time requirements of the solution of the NP-hard optimisation problem. In this context, the interaction of suitable heuristic optimisation algorithms and simple discrete queuing models is of particular interest. Given this context, the objective of the research presented in this paper is to investigate on the one hand the performance of the Cell Transmission Model (CTM) ((3) and (4)) compared to well known continuous queuing model, e.g. (1) and (10), and on the other hand to prove the applicability of the CTM for online control purposes.
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THE CELL TRANSMISSION MODEL As discretised version of the macroscopic modelling approach, the CTM has been developed by (2) and (3). The CTM provides a convergent approximation to the LWR model ((11) and (12)) and can be used to predict transient phenomena such as the build-up, propagation and dissipation of queues. The CTM employs a simplified version of fundamental diagram usually based on a trapezium form (see Figure 1) assuming that a free-flow speed vf at low densities and a backward wave speed -w for high densities are constant. Flow q
qmax q
vf
-w
v
Kjam
k
Density k
Figure 1: The Flow Density Relationship for the CTM The following 5 sections, which are considered as the basis of CTM, summarize the essential results of Daganzo’s work. After that the determination of delay and then new model considerations are explained.
Network model and topologies. In the CTM, the roads of the network are formed by the formation of successive sections or cells. The CTM uses the common modelling of traffic networks by directed graphs with nodes and links. In the system, the cells are described as the nodes {I} and the possible vehicle transfers by a set of links {k}. To define the topology of the network, a beginning (upstream) and an ending (downstream) cell for each link k is specified. The prefixes B and E are added to the link label, so that the beginning and ending cells of link k become BK and EK respectively. CTM limits the number of links connecting a cell to three. This results in the following three types of connections: • The simple or normal connection shown in Figure 2a, where one link enters a cell and one leaves it. • The diverge connection shown in Figure 2b, where one normal link enters a cell and two links diverge from it. • The merge connection shown in Figure 2c, where two links merge to a cell and one normal link leaves it.
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Bk
k
Ek
a) normal k
Ek
Bk
k Ek
Bk ck
ck
Ck
Ck
c) merge
b) diverge
Figure 2: Modelling of normal, diverge, and merge links
Traffic flow on normal link Discretising the road into homogeneous sections (or cells) and time into intervals of duration T such that the cell length L is equal to the distance travelled by free-flowing traffic in one time interval, the LWR results are approximated by a set of recursive equations:
n ' Bk (t + 1) = n Bk (t + 1) − q k (t ) ;
n ' Ek (t + 1) = nEk (t + 1) + qk (t )
q k (t ) = min{n Bk (t ), min[QBk (t ), QEk (t )], ( w / v f )( N Ek (t ) − n Ek (t ) )}
[1]
[2]
The variables n(t), N(t), q(t), and Q(t) denote the actual number of vehicles (occupancy), the maximum number of vehicles (or holding capacity), the actual inflow, and the inflow capacity that can be present in cell B or E at time t, respectively. With equation [2], the entire range of traffic densities described in Figure 1 of the simplified fundamental diagram is considered in the calculation of traffic flow. Based on the value of traffic density, three cases of traffic conditions are considered: • The first case occurs when the density is low (free flow condition), all vehicles can travel from cell BK to cell EK at time t. • The second case occurs when the traffic demand becomes high resulting in increasing of density, then the traffic flow is limited by the smaller value of the capacities QBK(t) or QEK(t). • The third case occurs when the density of the downstream cell is high, then not all of vehicles can travel from cell BK to cell EK. At this condition, the traffic flow is limited with the free space available (NEk(t)-nEk(t)) in the downstream cell EK. Equation [2] can be simplified by the following two equations:
S BK (t ) = min{QBK (t ), nBK (t )} and REK (t ) = min{QEK (t ),(w / v f )(NEK (t ) − nEK (t ))} [3]
q k (t ) = min{S Bk (t ), REk (t )}
[4]
where the actual flow on link k qk(t) is the minimum of sending SBK(t) and receiving SBK(t). ITS HANNOVER 2005
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After the flows have been determined for each cell for a specified time step, the occupancies can be easily updated with the conservation equation [1] by taking the link flows away from the beginning cells and adding them to the ending cells. The occupancies, n`(t+1), are intermediate variables introduced; they can be eliminated during computer operation.
Traffic flow on diverge links In Figure 2b, cell BK is the start of diverge to either cell EK or cell CK. The maximum number of vehicles SBk(t) that can be send from cell BK and the maximum number of vehicles (REk(t) and RCk(t) ) that can be received by cells EK and CK respectively are determined using equation [3]. The proportions of vehicles βEk and βCk leaving cell BK are assumed to be constant, where βEk+ βCk=1. Since part of flow leaving cell BK is diverging to cell CK and part to cell EK, all flow is impeded if either one of the diverging links is unable to accommodate its share of flow in order to maintain first-in-first-out queuing regime. This means that vehicles unable to exit will prevent all those behind, regardless of destination, to move. Considering these conditions, the flow that exits from BK can be determined by the following equation: q Bk (t ) = min{S Bk , REk / β Ek , RCk / β Ck }
[5]
The flows on both diverging links can be determined as follows: q k (t ) = β Ek q Bk und qCk (t ) = β Ck q Bk
[6]
Traffic flow on merge links As shown in Figure 2c, cells B and C are the start of a merge to cell E. SBK and SCK are the outflows (Sending) that can be send from cells B and C respectively. REk is the inflow (Receiving) ability of cell E. Depending on whether the receiving of BK is less or more than the sending of CK and EK, three cases must be distinguished to be able to build the equations for the flows on the two links: Case 1: If the receiving RE is more than the sending of the two cells SB + SC, then the actual outflows are determined as follows: for REk ≥ S Bk + S Ck :
q k (t ) = S Bk und qCk (t ) = S Ck
[7]
Case 2: If the receiving is less than the sending of the two cells, then the actual outflows are determined as follows: for S Bk > REk pk ∧ S Ck > REk pCk :
q k (t ) = p k REk and qCk (t ) = pCk REk
[8]
The constant pk is the proportion of vehicles comes from BK and the remainder pck from Ck, wher pk+pck=1, and pk/pck = qk/qck. These constants are the characteristics of merge junctions that describe any priority. Case 3: If the sending of one of two cells is limited by the receiving, then the actual outflows are determined as follows: for S Bk < REk p k ∧ S Ck > REk pCk :
q k (t ) = S Bk and qCk (t ) = REk − S Bk
[9]
for S Bk > REk pk ∧ S Ck < REk pCk :
q k (t ) = REk − S Ck and qCk (t ) = S Ck
[10]
For the cases 2 and 3, (3) has proved that on links k and ck can be determined by following simple equations:
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for
REk < S Bk + S Ck :
[11] q k (t ) = mid {S Bk , REk − S Ck , p k REk } qCk (t ) = mid {S Ck , REk − S Bk , pCk REk }
Traffic signal control The established characteristic quantities (Q, N, S, R, ß, p) were accepted as constant values till now, however, they also can be time variant. For example, a time variant capacity Q (t) can be used to model inflow controls. Traffic signal control can be shown by the restriction of the capacity in the controlling cells of the road junction or by the restriction of a time variant p(t). In this case p takes the value 0 or 1, depended whether the signal group is closed or released. The traffic flow equation is then for p k (t ) = 0 :
q k (t ) = 0
[12]
for p k (t ) = 1 :
q k (t ) = min{S Bk , REk }
[13]
With the time and place discrete representation of the traffic signal control, one can quantify the effect of the variation of all control variables (cycle time, split, green time and offset).
Delay Estimation In the flow density diagram shown in Figure 1, the slope of the line drawn from the origin represents the actual speed vI(t) in cell I at time t which is equal to the outflow divided by the density as follows: vI (t ) =
qk (t ) k I (t )
[14]
nI (t ) L
[15]
where: k I (t ) =
For simplicity, the relationship between the delay of one vehicle in cell I at time t and the actual speed vI(t) is assumed to be linear as shown in Figure 3: Delay dI(t)
T
vf
Speed vI (t)
Figure 3: The simplified delay-speed relationship
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Based on this relationship, delay ranges from 0 when the actual speed is equal to vf to the duration of T (time step) when the vehicles are not moving. This delay times the number of vehicles nI(t) determines the total delay of all vehicles in cell I at time t: ⎡ ⎛T ⎞⎤ d I (t ) = ⎢T − ⎜ × vI (t ) ⎟⎥ × nI (t ) ⎜ ⎟ ⎢⎣ ⎝ v f ⎠⎥⎦
[16]
where: vf =
L T
[17]
By substituting equations [14], [15], and [17] in equation [16], the following equation results: d I (t ) = T [nI (t ) − qk (t ) × T ]
[18]
Once delay has been determined at the cell level, the total delay Dlink can easily be estimated at the link level by summing up all delays in all cells during the cycle time as follows: link
Dlink = ∑∑ d I (t ) t
[19]
I
COMPARISON BETWEEN CTM AND WELL KNOWN QUEUING MODEL To evaluate the delay estimation based on the CTM, a virtual environment was prepared by means of microscopic simulation (AIMSUN2) and was used as reference measurements of total delay in each cycle time. By different scenarios of traffic demand, tests and comparisons to well-known models ( i.e (1) and (10) ) were performed. In defined intervals of 5 seconds for the CTM and 90 seconds (= cycle time) for the other models, the flow was counted by the detector. The total cyclic delay determined by the simulation, which was supposed to be the real one, was used for evaluation. The simulation time is at least 1 hour in all the scenarios. An example of graphical presentation for one scenario is presented in Figure 4 which shows the comparison of the total vehicular delays (veh.s/90s) for different delay estimation models and the corresponding degree of saturation (x). It is shown from Figure 4 that, Kimber-Hollis (KH) for estimating the total delay give good results for relatively low degree of saturation, however when the degree of saturation increases the method overestimates the reference value. This problem is not seen in CTM. In contrast, Akcelik´s method gives good results for relatively low degree of saturation, however when the degree of saturation increases the method underestimates the reference value. This problem is also not seen in CTM.
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1.6
7000
1.4 Real Kimber-Hollis Akcelik CTM x
6000 5000 4000 3000
1.2 1 0.8 0.6
2000
0.4
1000
0.2
x - degree of saturation [-]
Total delay [veh.s/90s]
8000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
0
intervals (90 s)
Figure 4: An example of the comparison of the total vehicular cyclic delay for the different queuing models.
ALGORITHMS FOR OFFSET OPTIMISATION As objective for the optimization, the minimisation of the total network delay was chosen: f = min ∑∑ d j (t ) t
[20]
j
where f is the sum of delays in all the cells throughout the planning horizon. The objective is to select the offset values for all intersections such that f is minimized. This minimization is subject to the constraint of the dynamic traffic demand. When one has fixed phase sequences, cycle times and green times for n signalized intersections in a road network, then one can have a number of ct(n-1) combinations (ct is the duration of the cycle time) for the solution space of offsets optimization. For small network, the optimal solution can be determined exactly with an already high computation time by complete enumeration. For larger networks and particularly for online use, this is no longer possible because of the required CPU time. Since the objective function has an irregular shape in the solution space and therefore the classical search methods cannot be employed, two heuristic approaches based on genetic algorithm (GA) will be used (7). In the two approaches, each decision variable (offset) is coded as a seven-bit binary string (chromosome), allowing for a total of 128 (27) distinctive values. For offset optimization, offset values are typically defined in the unit of seconds and are not higher than 120 s (cycle time). Therefore, seven-bit binary strings are sufficient for practical applications.
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The two methods used are different in the choice of the search direction. In the first approach, which is called here parallel genetic algorithm (PGA), a simultaneous search over all offset times by the process of variation of reproduction – crossover – mutation of the entire chromosome is performed. A convergence criterion or the computing time available ends the search. In the second approach, which is called serial genetic algorithm (SGA), only one offset value and therefore only a part of the chromosome is varied until the best solution is found. In the next step, the offset the next intersection is optimized. In such serial search, the order of treating the intersection and searching the offset has a great influence on the optimization results. For the following application examples, the search order has been fixed based on engineering practical experiences. For a further use of the SGA, a method should be found for the determination of the search order.
PERFORMANCE ANALYSIS IN LABORATORY TESTS For laboratory test, the CTM was converted prototypically in a C++ program. The simulation system AIMSUN2 serves as a virtual test environment, in which under constant boundary conditions, the delays can be estimated quantitatively depending on the offset times at signals. With this test environment, a hypothetical two-way arterial with three junctions as well as a small real road network with six road junctions were built. The considered time horizon is 900 s. This duration is considered for both the optimization by the CTM and the performance analysis in AIMSUN2.The traffic demand during this time period is assumed to be constant. The time is subdivided into discrete intervals of 1s in the CTM, whereby in connection with the free flow speed, the length of the cells was also fixed. As a comparison criterion for the quality of the optimization, the range of performance is traced out on one hand with a randomly selected set of offset values and on the other hand with an optimal set of offset determined by full enumeration. In addition, the offset values determined without using a computer program by the dominance method, see (13), are investigated, which is a known method for engineering practice. The reference offset values determined by the mention methods are compared with the values which were obtained by the optimization based on PGA and SGA.
Offset optimization in a hypothetical two-way arterial Figure 5 shows the CTM representation of the hypothetical two-way arterial with the required cells and links. The three signalized intersections have a common cycle time of 70 s, each intersection in the main direction has a 20 s green time and the inter-green time is 5 s. For the simulation of this network using the CTM with a time horizon of 910 s, one needs 400 ms on a PC (Athlon-1 GHz). For the optimization with the PGA, 25 generations were chosen with a population of 10 chromosomes, while with the SGA, 8 generations were chosen for each sequence with a population of 5 chromosomes.
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153 154
1
22
183
118
159
117
164
31
25
29
24
28
23
27
22
34
35
36
37
39
40
41
38
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49
22
73
74
75
76
82
83
77
78
79
43
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44
46 47 48
113 109
99
100
112 108
111 107
106
102
101
84 80 981
93
97
92
96
91
95
90 22
22
114 110
121
941
1
30 1
1
26 1
1
12
81
1
187
145
1
188
146
1 189
147
119
1
190
14 8
1
1 1
1
1
22
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1
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120
115
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116
1
151
152
167
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11
16
170
171
162
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175
1
10
172
1
9
173
1
15
176
161
11
8
14
1
22
13
177
174
1
2
1
1
185
178
165
1
181 179
186
160
180
22
182
1
163
22
158 184
22
1
1
86
18
85
17
Figure 5: CTM representation of the fictitious two-way arterial Table 1 summarizes the results of optimization of the offsets based on different algorithms. It can clearly be seen from the results that there is a significant potential for optimization of 40% between the worst and the best solution. With both PGA and SGA the best solution could be found, whereas the SGA could reduce the CPU-time from 100 s to 20 s. With the dominance method one can obtain reasonable results, however, the absolute optimum could not be found.
Method Exact optimum by full enumeration Random bad coordination Dominance method PGA SGA
Offsets at intersections 1 2 3
Total delay Veh.s
0
1
37
14872
0.0%
15 min
0
58
46
20878
40%
15 min
0
8
40
16445
10.6%
1s
0
1
37
14872
0.0%
100s
0
1
37
14872
0.0%
30s
Relative changes %
CPU time
Table 1: summary of results of performance analysis for the fictitious two-way arterial
Offset optimization in a small road network Figure 6 shows the CTM representation of a small network that is a part of List district in Hannover city. The 6 signalized intersections of the network have a common cycle time of 90s. For the simulation of this network using the CTM with a time horizon of 900s, one needs 700ms on a PC (Athlon- 1 GHz). For the optimization with the PGA, 20 generations were chosen with a population of 50 chromosomes, while with the SGA, 10 generations were chosen for each sequence with a population of 5 chromosomes.
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K2
K6
K4 20
16
17
15
9 14
12
11
7
10
19 8 13
6
4
5
3
18
2
1
0
K1
K3
K5
0
0
Figure 6: CTM representation of the grid network Table 2 summarizes the results of the offset optimization with the different methods. The results show that for the real network, there is a potential for improvement of about 20% less delay than the existing coordination plan. With both the PGA and the SGA, a solution close to the absolute optimum could be found. Intersection
1
2
3
4
5
6
Existing Plan Worst plan Enumeration Algorithm GA SGA
0 40 87 0 87
7 0 82 81 84
0 70 14 18 9
3 0 0 0 0
83 40 48 33 40
86 80 88 21 0
Total delay Veh.s 49214.3 90704.5 40695.9 42180.6 41191.6
% increase 20.9% 122.9% 0.0% 3.6% 1.2%
CPU time
24 hr 700 s 175 s
Table 2: Summary of results of performance analysis for the realistic network Because of the low interdependencies within the chosen network, the SGA performs as well as the PGA. However, it is assumed that with increasing multiple dependencies of the offsets like in the case of larger grid network, this will not be true. The duration of the optimization is influenced by the simulation time, which increases linearly with the network size, and by the complexity of the optimization problem, which increase exponentially. The results with regard to the required CPU time make clear that the limits of online adaptation are already reached for the comparatively small test network.
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CONCLUSION Offset optimization in networks represents a special difficulty since the discrete variables can not be represented like the cycle time or the green time in the form of the relationship values and therefore cannot be modelled by efficient means of queuing theory. For the assessment of the single solutions in the optimization process, one needs the time and space discrete traffic flow models which must obtain very short calculation time because of the complexity of the optimization task. For this purpose, the cell transmission model (CTM) provides a new approach with minimal computing time requirements, which in connection with heuristic optimization algorithms is presented in this paper. The known model equations are introduced and extended by a formulation of the vehicular delay. It is shown in empirical examinations that the real flow of traffic and the performance criteria like delay and queue lengths can be well simulated with the CTM. As optimization algorithms, two approaches are introduced on the basis of genetic algorithms. Besides the known approach, which encodes all variable values in a chromosome and modifies the values in parallel, a serial procedure was introduced which optimizes variable by variable successively. This serial genetic algorithm is a first attempt to reduce the required optimization time, however, it requires further considerations for the determination of the order. The potential of the introduced optimization methods was checked with a short hypothetical two-way arterial as well as with a small real network. It could clearly be shown that delay savings are obtainable, whereby the calculation speed on normal PCs permits an online use for small networks.
ACKNOWLEDGEMENTS The software for this work used the C++ GAlib (genetic algorithm library), written by Matthew Wall at the Massachusetts Institute of Technology.
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5. Donati, F.; Mauro, V., Roncolini, G. and Vallauri, M. (1984): “A Hierarchical Decentralized Traffic Light Control System,” The First Realisation: ‘Progetto Torino’. In: IFAC 9th World Congress, Vol II, 11G/A-1. 6. Friedrich, B. and Keller, H. (1994): “BALANCE - A Method for Integrated Traffic Adaptive and Vehicle Actuated Signal Control,” In: Proc. of the 7th IFAC Symposium, August 24-26, 1994, Tianjin China. 7. Goldsberg, D. E. (1989): “Genetic algorithms in Search, Optimization, and Machine Learning,” Addison-Wesley Publiching Company, INC.. 8. Henry, J. J., Farges, J. L. and Tufal, J. (1983): “The PRODYN real time traffic algorithm,” In: Proc. of the IFAC Symposium, Baden-Baden. 9. Hunt, P. B., Robertson, D. I., Bretherton, R. D. and Winton, R. I. (1981): “ SCOOT - a traffic responsive method of coordinating signals,” TRRL Laboratory Report 1014. 10. Kimber, R. M. and Hollis, E. M. (1979): “Traffic queues and delays at road junctions,” TRRL Laboratory Report 909. 11. Lighthill, m.j. and Whitham, j.b. (1955): “On kinematic waves. I. Flow movement in long rivers. II. A theory of traffic flow on long crowded road,” Proceedings of Royal Society,A229, 281-345. 12. Richards, P.I. (1956): “Shockwaves on the highway,” Operations Research B 22, 81-101. 13. Schnabel, W. (1981):“Verkehrstechnische Berechnung von lichtsignalgesteuerten Straßennetzen,“ Forschungsbericht, Zentrales Forschungsinstitut des Verkehrswesen der DDR, Berlin.
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