### Traffic Signal Optimization

Project director: |
Prof. Dr. Rolf H. Möhring |

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany | |

Tel: +49 (0)30 - 314 24594 | |

e-mail: moehring[at]math.tu-berlin.de | |

Researcher: |
Gregor Wünsch |

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany | |

Tel: +49 (0)30 - 314 23598 | |

e-mail: wuensch[at]math.tu-berlin.de | |

Support: |
Bundesministerium für Bildung und Forschung (BMBF) |

DFG Graduiertenkolleg GrK 621-2: | |

Stochastische Modellierung und quantitative Analyse großer Systeme in den Ingenieurwissenschaften (MAGSI) | |

Cooperation partner: |
Planung Transport Verkehr AG (PTV) |

### Problem description.

Todays traffic situations in urban areas require intelligent traffic signal control. Many of the traffic signal controlling strategies, which are in pratical use today, only consider every intersection individually. This surely does not lead to a good network-wide controlling or coordination, respectively, since no synchronization aspects as green waves are recorded in the models. Our approach goes beyond this limitation. We develop an optimization model, namely a mixed-integer linear program, that minimizes delays in an signalized inner-city traffic network. This is done by, e.g. creating the above mentioned green waves. It has to be mentioned that we restrict to fixed-time controlled signals, which is motivated by the fact that we assume a constantly high traffic volume. The model's decision variables are the offsets between the signalized intersections.

#### Application.

Our cooperation partner, the ptv AG from Karlsruhe, Germany, is one of Europe's leading traffic software developing companies. Once our optimization model becomes competitive, it will be suitable for the ptv software. Presumably, it best fits in VISUM, a macroscopic transportation planning and travel demand modeling tool. Inside VISUM our tool shall close the gap that arises, when the programs' dynamic traffic assignment includes the networks' signals' offsets for calculating the user equilibrium.

### Model.

We decided to build up a mixed-integer linear program (MIP) minimizing the total occurring delay of the vehicles in the network. Therefore we represent the road network as a directed graph and introduce convex link performance functions (LPFs) for each arc, see for example[1]. The fix traffic flow is modeled macroscopicly, which means that we look at streams, or platoons, of traffic instead of singular vehicles. So, according to the platoon's head's arrival time, which is fixed by the link's offset and the adjacent signals' phase sequences, the lost time due to poor synchronization of the signals is measured. In doing so, the LPF's domain has to have a length of size of the adjacent signals' greatest common divisor, in order to save additional integer variables.

Furthermore, phase sequencing selecting constraints and network topological loop equations have to be defined. Whereas the former ones are simple to model, just by introducing binary variables selecting one of several predetermined modes, the latter ones are somewhat harder to establish in the case of non-uniform cycle times. Questions concerning so called loop cycle times and cycle bases arise, see e.g. [5], that come from the field of periodic event scheduling problems [4].

#### Evaluation.

At the end of a new optimization tools' development surely an evaluation is needed. Therefor the optimization results are simulated in ptv's microsimulation tool VISSIM, where we measure several performance measures, such as waiting time or the number of stopps. But since it is hard to value the results by their own, a comparison is necessary. So, a solution, namely a set of offsets, is extracted from hill-climbing and genetic algorithm approaches done by Transyt[3], a tool widely used mostly in the United States.

### References.

[1] | N.H.Gartner, J.D.C.Little, H.Gabbay.
Optimization of Traffic Signal Settings by Mixed Integer Linear Programming.
Transportation Science 9, pp.321--363, 1975. |

[2] | C.F.Daganzo.
Fundamentals of Transport and Traffic Operations.
Elsevier Ldt., 2000. |

[3] | D.I.Robertson.
Transyt: a traffic network study tool.
RRL Report LR 253, 1969. |

[4] | P. Serafini, W. Ukovich.
A Mathematical Model for Periodic Scheduling Problems.
SIAM Journal of Discrete Mathematics Vol.2, No.4, pp.550--581, 1989. |

[5] | S. Haenelt
Taktfahrplanoptimierung mit unterschiedlichen Taktzeiten.
Masters' Thesis, TU-Berlin, 2004. |

#### See also:

**Publications**R.H.Möhring, K. Nökel, G. Wünsch:*A model and fast optimization method for signal coordination in a network*, In: van Zuylen, H., Middelham, F., Hrsg.: Proceedings of the 11th Symposium on Control in Transportation Systems - CTS 2006, S. 73-78, Delft, Niederlande 2006.