Study of Correlation Among Several Traffic
Parameters Using Evolutionary Algorithms:
Traffic Flow, Greenhouse Emissions and
Network Occupancy
Javier Sánchez Medina, Manuel Galán Moreno, and Enrique Rubio Royo
Innovation Center for Information Society – C.I.C.E.I.,
University of Las Palmas de Gran Canaria,
Campus de Tafira s/n, Las Palmas de Gran Canaria, Spain
[email protected], [email protected], [email protected]
http://www.cicei.com Abstract. During the last two years we have been working on the optimisation of traffic lights cycles. We designed an evolutionary, distributed architecture to do this. This architecture includes a Genetic Algorithm for the optimisation. So far we have performed a single criterion
optimisation – the total volume of vehicles that left the network once
the simulation finishes. Our aim is to extend our architecture towards a
multicriteria optimisation. We are considering Network Occupancy and
Greenhouse Emissions as suitable candidates for our purpose. Throughout this work we will share a statistical based study about the two new
criteria that will help us to decide whether to include them or not in the
fitness function of our system. To do so we have used data from two real
world traffic networks.
Keywords: Genetic Algorithms, Cellular Automata, Traffic Optimisation, Greenhouse Emission, Multicriteria Optimisation.
1
Introduction
The traffic issue means, for every major city in the world, not only a quality and
comfort factor, but also a very important economic subject.
The non-stopping overload process that traffic infrastructures are suffering
calls for new solutions. Moreover, the world governments are aware about the
global warming risks. Hence, it is a must to optimise the existing infrastructures
to obtain the very best service they can provide, minimising their environmental
impact.
In our group we have been working on the optimisation of traffic lights cycles
for the better performance of urban traffic networks. As shown in [1], traffic light
This research have been partially supported by the Ministerio de Educación y Ciencia
of Spain (ref TSI2004-05949), which is financed 70% from European Union FEDER
funds.
R. Moreno-Dı́az et al. (Eds.): EUROCAST 2007, LNCS 4739, pp. 1134–1141, 2007.
c Springer-Verlag Berlin Heidelberg 2007
Study of Correlation Among Several Traffic Parameters
1135
cycles have a strong influence in traffic flow results. We have designed a three
pillar model consisting of a Genetic Algorithm (GA) as optimisation technique, a
traffic microscopic simulator inside the fitness function of the GA, and a Beowulf
Cluster as scalable MIMD multicomputer.
We have been testing this architecture through two years with successful results. So far we have used a single criterion for the fitness function of the Genetic
Algorithm – the total number of vehicles that left the network once the simulation finishes.
Currently, our aim is to extend this system including other criteria. We are
now focused on the occupancy of the network and the greenhouse emissions as
new criteria candidates. Across this paper we present a statistical study comparing the three criteria, and using two real world networks as test cases. The
rest of the article is organised as follows. In the next section we describe our
methodology. In section 3 we present the new criteria considered. In section 4
we show the tests performed and the obtained results. Finally, some concluding
remarks and future work ideas are presented in section 5.
2
Methodology Description
The architecture of our system comprises three items, namely a Genetic Algorithm (GA) as Non-Deterministic Optimisation Technique, a Cellular Automata
(CA) based Traffic Simulator inside the evaluation routine of the GA, and a
Beowulf Cluster as MIMD multicomputer. Through this section we will give a
succinct description for the GA and the CA Traffic Simulator used. Finally, a
brief description of the Beowulf Cluster will also be given.
2.1
Genetic Algorithm
– Fitness Function. After testing several criteria we found out that we obtained
the best results by using the absolute number of vehicles that left the traffic
network during the simulation as criterion.
– Chromosome Encoding. A Gray Code encoding ( [2]) is used to codify the
length (seconds) of every one of the states associated to an intersection. The
states sequence is preestablished.
– Selection Strategy. We have chosen a Truncation and Elitism combination
as selection strategy. This combination seems to be the most fitted to our
problem among a set of selection tested strategies.
– Crossover Operator. We use a standard Two Points Crossover.
– Mutation Operator. The value of a randomly chosen bit in the chromosome is
just flipped. The mutation probability is variable, starting with a very high
probability that decreases generation by generation.
2.2
Traffic Simulator
Inspired on the Cellular Automata Model we have developed a non-linear model
for simulating traffic behaviour. We have developed a traffic model based on the
1136
J. Sánchez Medina, M. Galán Moreno, and E. Rubio Royo
SK1 model ( [3]) and the SchCh2 model ( [4]). The SchCh model is a combination
of a highway traffic model — Nagel-Schreckenberg ( [5]) — and a very simple
city traffic model — Biham-Middleton-Levine ( [6]). The SK model adds the
“smooth braking” for avoiding abrupt speed changes. We decided to modify our
model inspired by the SK model due to its improved results for all the tests
shown in Brockfeld et al. ( [7]).
2.3
Beowulf Cluster
The Architecture of our system is based on a five node Beowulf Cluster, due to
its very interesting price/performance relationship and the possibility of running
Open Software on it. On the other hand, this is a very scalable MIMD computer, a
very desirable feature in order to solve all sort — and scales — of traffic problems.
3
Optimisation Criteria Under Study
3.1
Traffic Flow
As explained in section 2, in our methodology we use the total volume of vehicles
that left the network once the traffic simulation finishes as fitness value. In order
to compare that parameter with the Total Emissions and Occupancy relative
parameters – subsections 3.2 and 3.3 – it is required to derived a new relative
parameter. So, for this current study we define the Exit Probability (equation 1).
The Exit Probability is the number of vehicles that left – Nvl – over the number
of vehicles that entered the network – Nvi – once the simulation finishes.
Exit Probability =
3.2
Nvl
Nvi
(1)
TE =
Ncells
−1
(fEF (i, c))
(2)
c=0
Total Emissions Criterion
In [8] it is shown how CO and N Ox are mostly linearly correlated with the
speed of vehicles. This fact may be clearly observed in pictures 4 and 5 of that
work. With this in mind we define a new parameter, Emission Factor, as the
scalar value of the speed of every vehicle, every simulation iteration. This such
low computing consuming parameter will give us an approximated idea of the
volume of greenhouse gases emitted during the simulation.
In our tests we registered iteration by iteration the global sum of emissions –
Total Emission (TE) – as shown in equation 2. In this equation Ncells means the
number of cells in the traffic network; and fEF means the Emission Factor of
the vehicle at cell ’c’ and iteration ’i’. Obviously, when the cell is not occupied,
the fEF value is set to 0.
We believe that the minimisation of TE would mean the minimisation of
emissions, and we want to test if it is worth to be included in the fitness function
1
2
Stephan Krauss, the author.
Andreas Schadschneider and Debashish Chowdhury, the authors.
Study of Correlation Among Several Traffic Parameters
1137
or not. Throughout this study we have used the Mean and Standard Deviation
of TE calculated across all the simulation iterations.
3.3
Occupancy Criterion
The second criterion we have considered is the Occupancy. During the optimisation we obtain the average “State of Congestion”(SOC) and the average “Time
of Occupancy” (TOC). SOC was defined in [9]. TOC was defined in [10]. In
equations 3 and 4 we represent both parameters. About equation 3, Nco means
the number of cells occupied by a vehicle, and NcT means the total number of
cells in the treated network. About equation 4, Nito means the number of simulation iterations that a particular cell is occupied by any vehicle, and NitT means
the whole number of iterations that the simulation lasts.
As one may infer from equations 5 and 6, the average SOC across all the
simulation iterations and the average TOC across all the cells in the network
are the same value. In other words, the mean value of the average occupied cell
ratio across all the simulation iterations is equivalent to the mean value of the
average number of occupied iterations for a particular cell across all the cells in
the traffic network.
SOC =
Nco
NcT
SOC =
T OC =
(3)
NitT
Nco (i)
NcT
T
Nit
i=0
NitT
=
NcT
T OC =
o
Nit
(c)
T
Nit
NcT
c=0
o
i=0 Nc (i)
NcT ∗ NitT
Nito
NitT
(4)
(5)
NcT
=
o
c=0 Nit (c)
T
Nc ∗ NitT
(6)
In this research, we have considered two different moments for the Occupancy.
They are the average and the standard deviation of SOC values measured at
every iteration across the whole simulation.
4
4.1
Statistical Study
Test Results
We have used two traffic networks for this research – Figures 1 and 2. The first
one is situated in “Las Ramblas” – Santa Cruz de Tenerife, Canary Islands,
Spain. The second one is situated in “La Almozara”, in the city of Zaragoza,
Spain. The size of the network was 1643 cells for “Las Ramblas” and 2753 cells
for “La Almozara”.
We have used a population of 200 individuals, and we have run the Genetic
Algorithm through 200 generations. For each test case we have run 30 executions
of the Genetic Algorithm. The traffic simulations run during 2000 iterations for
“Las Ramblas” and 4000 iterations for “La Almozara”.
1138
J. Sánchez Medina, M. Galán Moreno, and E. Rubio Royo
Fig. 1. “Las Ramblas” zone
Fig. 2. “La Almozara” zone
Once the 60 executions of the GA finished we picked up the following parameters from the final populations: Total Flow (Fitness), Occupancy and Total
Emissions. Our aim is to see how correlated are they. As we said in 3.3, every
iteration the average and the standard deviation of SOC were measured. We
sampled the average and standard deviation of Total Emissions too.
In figures 3, 4, 5, 6, 7 and 8 we have represented 4 pairs of parameters for
“Las Ramblas” and 2 pairs for “La Almozara”. In these pictures we have also
represented the linear regression function using the LMS fitting algorithm at
each subplot. For that purpose we have used the polyfit and polyval functions
of Matlab. Finally, we have also calculated the Pearson Correlation Coefficient
(equation 7) for each case.
n
(xi − x)(yi − y)
n
(7)
ρ = n i=1
2
(x
−
x)2
i=1 i
i=1 (yi − y)
Study of Correlation Among Several Traffic Parameters
0.25
1139
420
0.24
400
rho = −0.62
0.23
Rho = −0.67
y=−0.63x + 0.74
y=−873.39x + 1077.02
380
Mean TE
Mean SOC
0.22
0.21
360
340
0.2
0.19
320
0.18
300
0.17
0.8
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
280
0.8
Exit Probability
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
Exit Probability
Fig. 3. Las Ramblas – Exit Probability Fig. 4. Las Ramblas – Exit Probability
Versus Mean SOC
Versus Mean Total Emissions
420
400
130
Rho=0.99
120
y=1278.54x + 80.26
Rho = 0.98
380
y= 1342.55x + 7.06
360
Std TE
Mean TE
110
100
340
90
320
80
300
280
0.17
0.18
0.19
0.2
0.21
Mean SOC
0.22
0.23
0.24
0.25
70
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
Std SOC
Fig. 5. Las Ramblas – Mean SOC Versus Fig. 6. Las Ramblas – Std. SOC Versus
Mean Total Emissions
Std. Total Emissions
4.2
Discussion
First of all, one may see that there exists a strong linear relationship between
the mean and std. SOC and the mean and std. Total Emissions respectively –
figures 5,6, 7 and 8. This fact is specially clear in figure 7, where the Pearson
coefficient is 0.99.
It seems clear that the Occupancy and Total Emissions criteria are very related. This fact conforms with reality since a very loaded traffic network is also
likely to be a very pollutant one, and vice versa.
Hence, we can not use the two criteria simultaneously in a hypothetical multicriteria fitness function. We can either minimise the emissions while maximising
the occupancy or the reverse. Thus, we will have to choose between the minimisation of emissions or the maximisation of the occupancy.
1140
J. Sánchez Medina, M. Galán Moreno, and E. Rubio Royo
22
45
20
40
Rho = 0.99
Rho = 0.98
18
y=2669.79x + 8.39
y = 2341.33x + 2.84
16
Std TE
Mean TE
35
30
14
12
10
25
8
20
6
15
2
4
6
8
Mean SOC
10
12
14
−3
x 10
4
1
2
3
4
5
Std SOC
6
7
8
−3
x 10
Fig. 7. La Almozara – Mean SOC Versus Fig. 8. La Almozara – Std. SOC Versus
Std. Total Emissions
Mean Total Emissions
Secondly, as may be observed in figure 3, there exist a clear relationship
between the Exit Probability and mean SOC. This is not a linear relationship,
but a higher order one. In further research we will explore this.
Also from figure 3 is observed that the Exit Probability and mean SOC are
negatively associated. This is consistent with common sense, since a congested
network is likely to have a small Exit Probability values and vice versa.
Figure 4 shows another case of nonlinear negatively associated relationship
between the Exit Probability and Emissions. Again, this makes sense since experience says that a very pollutant network – for instance a congested network
– is likely to have a low Exit Probability value.
5
Conclusions
Throughout this paper we have presented a study comparing several parameters
for developing a multicriteria fitness function, extending the consolidated traffic
light optimisation methodology we designed in our group.
In this work we have tested several criteria related to network occupancy and
greenhouse gasses emissions using two real world traffic networks: “Las Ramblas”
in Santa Cruz de Tenerife, and “La Almozara” in Zaragoza – Spain.
The results are enlightening and may be summarised as follows. First we
have found a strong linear and positive correlation between the occupancy of
the network and the emissions criteria. This is an important new information.
Because of it, in a hypothetical multicriteria fitness function we will have to
choose between maximising the occupancy of the network or minimising the
emissions.
We have observed nonlinear relationships between occupancy and the Exit
Probability too. A similar effect was observed when comparing the Exit Probability and Total Emissions.
Study of Correlation Among Several Traffic Parameters
1141
For future work we plan to determine the degree of the nonlinear relationships
found in this research. We are also planning to test new criteria. Once chosen
suitable criteria we will be able to define a new multicriteria fitness function for
our methodology.
Acknowledgements
The authors would like to acknowledge the Santa Cruz de Tenerife and Zaragoza
City Council Traffic Departments for their kind help. We specially would like
to thank Mr. Hilario Rodrı́guez González from Santa Cruz de Tenerife for his
willingness to collaborate with us, making this work feasible.
We also would like to thank Professor Marı́a Luisa Sein-Echaluce Lacleta,
from the University of Zaragoza for her kind help to obtain the data from the
Zaragoza Traffic Department.
References
1. Brockfeld, E., Barlovic, R., Schadschneider, A., Schreckenberg, M.: Optimizing Traffic Lights in a Cellular Automaton Model for City Traffic (2001),
http://arxiv.org/ps/cond-mat/0107056
2. Black, P.E.: ”Gray Code”, from Dictionary of Algorithms and Data Structures. In:
Black, P.E. (ed.) NIST, http://www.nist.gov/dads/HTML/graycode.html
3. Krauss, S., Wagner, P., Gawron, C.: Metastable states in a microscopic model of
traffic flow. Phys. Rev. E 55, 5597–5605 (1997)
4. Schadschneider, A., Chowdhury, D., Brockfeld, E., Klauck, K., Santen, L., Zittartz,
J.: A new cellular automata model for city traffic. In: Traffic and Granular Flow
1999: Social, Traffic, and Granular Dynamics, Springer, Berlin (1999)
5. Nagel, K., Schreckenberg, M.: A Cellular Automaton Model for Freeway Traffic.
Journal de Physique I France 33(2), 2221–2229 (1992)
6. Biham, O., Middleton, A.A., Levine, D.: Phys. Rev. A 46, 6124 (1992)
7. Brockfeld, E., Khne, R.D., Wagner, P.: Towards Benchmarking Microscopic Traffic
Flow Models Networks for Mobility. In: International Symposium, vol. I, pp. 321–
331 (2002)
8. Kean, A.J., Harley, R.A., Kendall, G.R.: Effects of Vehicle Speed and Engine Load
on Motor Vehicle Emissions. Environmental Science and Technology 37(17), 3739–
3746 (2003)
9. Bham, G.H., Benekohal, R.F.: A high fidelity traffic simulation model based on
cellular automata and car-following concepts. In: Transportation Research Part C,
vol. 12, pp. 1–32. Elsevier, Amsterdam (2004)
10. May, A.D.: Traffic Flow Fundamentals, vol. 78. Prentice-Hall, Englewood Cliffs
(1990)