Equilibria (equilibrium states) problem statement for city transport systems
Authors: Agureev I. E. , Akhromeshin A. V. , Pyshnyi V. A. 
Published: March 2023

Introduction. The article formulates the problems statement about equilibrium states in the model of a generalized transport system of the city, consisting of a street and road network, centers of mass gravity, places of residence of people, vehicles, as well as road users themselves, including passengers. The solution of these problems makes it possible to identify the equilibrium distributions of the system elements over various subsets of states (a subset of the elements of the road network; a subset of the center of mass gravity; a subset of trips of a certain type, etc.), depending on the type of vehicle, individual preferences, knowledge about the state of the transport system and other factors.At the same time, the transport system is considered as an object of research within the framework of the theory of macrosystems. A set of problems statements on the search for equilibrium states of the transport system for various modelling objects has been compiled for various structural levels (scales) of the objects under consideration.Materials and methods. In this paper, the theory of transport macrosystems is applied, which follows from a wellknown scientific discipline - the theory of macrosystems. Among its tasks there are statements about the distribution of elements over subsets of states and problems about the equilibrium of the system as a whole. In macroscopic systems, by definition, the stochastic behavior of a large number of elements transforms the deterministic behavior of the system as a whole. A macro system is thus a dynamic converter of the chaotic behavior of elements into a certain set of behavior parameters (phase variables) forming a space of small dimension. Therefore, within the framework of the theory of macrosystems, the basic concepts of entropy maximization at equilibrium states of the system are used. In this case, the distribution function of macrostates is selected depending on the method of filling some states with elements from the corresponding subsets; the necessary values of a priori probabilities and proofs of parametric properties of models of macrosystems with various statistics (Fermi-, Einstein- and Boltzmann-distributions). On the basis of the theory of macrosystems, for example, problems are solved to find equilibrium in such systems as: 1) megapolis with its functional and spatial structures (probabilistic states of hierarchical systems), 2) transport networks of cities formed by the movement of vehicles and residents of the city between different areas (distribution of trips along routes in the network); 3) logistics systems in the interregional exchange of products (problems of economic equilibrium in the exchange of resources).Results. The paper presents the results of research concerning the uniform description of the elements of the road network and the centers of mass gravity as components of the general transport system of the city (agglomeration) in the framework of the theory of transport macrosystems. At the same time, the study identifies various structural levels of description that can be used to solve particular problems, for example, finding equilibrium in individual subsets of the transport system, such as groups of centers of mass gravity of a certain type, or traffic flows on routes, stretches, network sections, etc.Discussion and conclusions. Within the framework of the work, the following tasks were solved: a description of the structural levels of the objects of the road network and the centers of mass gravity as the main components of the model of transport systems was developed; the formulation of problems about the equilibrium states of transport systems at the corresponding structural levels was developed; the analysis of the obtained methodology was performed; a methodological analogy is established between different subsets of states at the same structural level, for example, between the centers of mass gravity and elements of the road network as objects of modeling by methods of the theory of macrosystems (this analogy can be extended to other subsets of states in transport systems, for example, types of transport systems, travel purposes, parking spaces, subsystems of the intelligent transport system and much more).