Mathematicians from St Petersburg University find out how to get rid of traffic jams in large cities

Due to the fact that privately-owned vehicles all over the world are becoming more affordable, a problem often arises in large cities – vehicles cannot move freely. Scientists have been searching for a solution to this problem for a long time. Since the late 1950s, the theory of traffic flows has turned into a separate branch of applied mathematics. It is in recent decades that the relevance of such studies has grown several times.

‘In Russia, transport engineers historically take on the problem of traffic management,’ said Alexander Krylatov. ‘At the same time, they specialise more in solutions related to design changes of particular sections of the network. They do not have competencies in the field of system-related increases in traffic performance. With ever-increasing traffic flows, even if engineers manage to achieve local improvements, after a while the flows rearrange and the same traffic jams appear in other places.’

With ever-increasing traffic flows, even if engineers manage to achieve local improvements, after a while the flows rearrange and the same traffic jams appear in other places.

Alexander Krylatov, Professor at the Department of Mathematical Modelling of Energy Systems at St Petersburg University, Doctor of Physics and Mathematics

Alexander Krylatov together with Victor Zakharov, Professor at the Department of Mathematical Modelling of Energy Systems at St Petersburg University and Doctor of Physics and Mathematics, have written a monograph. It presents new mathematical approaches to traffic optimisation, as well as possible ways to implement them. The mathematicians’ monograph was published by Springer, an international publishing house.

The principles that the scientists recommend using as guidelines were formulated in 1952 by John Glen Wardrop, an English mathematician and transport analyst. The first of them – the principle of equilibrium – is a mathematical construct that makes it possible to simulate systems, in particular traffic, assuming that each driver pursues only their personal goals. That is why the models created with its help are based on the fact that behind any changes in traffic flows there should be the selfish behaviour of car owners.

The second principle – the Wardrop system optimal – states that there is the possibility of directive management of all vehicles. However, the authors of the monograph give specific importance to the first principle. They believe that the drivers’ behaviour can be influenced indirectly – through a change in road infrastructure. Mathematical models make it possible to predict how this will change traffic on each local section of the network.

The authors note that the drivers’ navigation systems have a great influence on traffic flow management. In their opinion, the most effective situation will emerge if all drivers use the same system and receive information on suitable routes from a single centre. Otherwise, if one of the major navigators announces without warning that it will redirect its users so that the traffic situation in the city improves and the other navigators do not support it, the changes will still remain at the local level – the system will readjust and the problem will not be solved.

Traffic optimisation is also possible using roadway widening or narrowing, which is especially important in cities with an already existing network. In such cases, it is often impossible to extend the road from one intersection to another one. Moreover, the construction of road junctions is not always efficient.

Where the road is difficult to physically widen, it is efficient to use other methods such as a ban on parking throughout the route. Additionally, science can help create dedicated lanes for electric transport if the city administration wants to motivate drivers to switch to green cars. It is possible to create separate special routes for them and it will be much easier to get around using them.

‘Every year, a considerable budget is allocated for improving roads. The mathematical theory of traffic assignment suggests a set of solutions for the efficient management of these funds,’ the scientist said. ‘The mathematical approach in this case is superior to the engineering and economic one. It makes it possible to analyse the entire transport network, with respect to the complex laws of the mutual influence of its individual elements on each other. We have done a lot of work in the field of simulating traffic flows and networks. Now we want to pass on to the stage of putting our ideas into practice.’  

One of the ways to use mathematical models can be the development of digital twins of transport systems based on these models. These simulations, implemented in the form of software applications, will become an extremely useful thought tool in the hands of transport engineers.

‘By building digital twins of the transport system and using them to optimise flows, it will be possible to achieve a balance between the demand for using the system and the infrastructure capabilities. It is unlikely that this can be done without economic digitalisation,’ added Victor Zakharov.