Multi-agent based optimal equilibrium selection with resilience constraints for traffic flow☆
Introduction
With the deepening of the modernization process, cities are becoming fragile and growing. When facing and encountering uncertain risks and disasters, the destruction or degradation of any part of the urban system will bring fatal threats to the whole city. More and more researchers began to consider the ability of transportation networks to resist and recover from disturbances, commonly known as resilience, in the planning and operation of transportation systems. The original meaning of resilience refers to the inherent characteristics of a spring, which can generally be understood as the stability characteristic of a material resisting external impact (Adger, 2000). The development of system theory extends the concept of resilience to other aspects, many of which lay a theoretical foundation for the study of network resilience in different research fields (Cheng et al., 2021, Modgil et al., 2021, Wu and Wang, 2021, Ye et al., 2021, Younis et al., 2021, Zhou et al., 2020). Roughly speaking, resilience refers to the ability of a system or community to adjust its activities to maintain an acceptable level of service in the face of errors, failures, and environmental changes. The transportation system is one of the crucial infrastructure systems, its smooth operation is of great significance to the entire transportation network. In particular, once some primary intersections or roads in the traffic network are seriously blocked, the operation of the whole network will be significantly affected. For current studies, reducing delays to improve the operation efficiency of the urban road systems is usually considered one of the optimization goals (Ali et al., 2022, Arcos-García et al., 2018, Cao et al., 2018).
Faced with man-made or natural disasters, it is a challenging question whether the traffic network can be restored to a new stable state rather than a state of paralysis. Fortunately, flow networks provide a practical model framework for such topics. Research on flow networks spans several research areas, including applied mathematics, computer science, engineering, management, and operations research (Huang et al., 2017, Wang et al., 2021). Indeed, typical questions of flow networks that have been studied widely include (Bertsekas, 1992): (i) Shortest path problem Find an optimal way to get from one point to another across the network at the lowest cost. (ii) Maximum flow problem. Finding a feasible flow through the traffic network to obtain the maximum possible flow rate. (iii) Minimum cost flow problem. Study how to transport goods residing at one or more points in the network to another or more points at the lowest possible cost. The problem discussed in this paper is related to the above minimum cost flow problem and can therefore be considered a potential application of the flow network problem.
For a dynamic traffic network, to study how to efficiently utilize road network resources (i.e., achieve the ideal network performance) with the minimum cost, we consider the traffic network framework proposed in Como et al. (2012) and design a distributed optimization algorithm. Indeed, the target of the studied problem is to determine the optimal equilibrium flow over a traffic network, assuming the demands from the source nodes to the destination node can be fully transferred (Ba et al., 2015). We introduce two critical concepts in our problem, strong resilience, and feasible equilibrium flow, thus constructing a convex optimization problem with the objective function of the general minimum cost. In our case, the strong resilience of the flow network is defined as the infimum sum of link-wise flow capacity reductions. It is consistent with the minimal residual node capacity if the drivers follow the locally responsive distributed routing policies (Como et al., 2012). We consider resilience constraints in our problem to ensure that the destination node’s total inflow asymptotically approaches the origin node’s total outflow. In addition, we use feasible equilibrium flow to describe the flow conservation and link capacity constraints at each node/intersection. The proposed problem’s feasible region is a convex set through these two concepts. Because the objective function of the optimization problem designed is strongly convex, it is reasonable to refer to the studied problem as an extension of the optimal equilibrium selection convex optimization problem. In addition, for the optimal equilibrium flow obtained by the design algorithm, we can generally induce the flow in the traffic network through traffic signal control to achieve flow equilibrium, for instance, intelligent road toll system (Sharon, 2021), manual traffic control (Parr & Wolshon, 2016), etc.
Based on the above discussions, we give the contributions of this paper as follows. Firstly, the existing research on traffic flow networks mainly includes network feedback control, network interconnection system, and network routing strategy (Como et al., 2014, Como et al., 2011, Como et al., 2012, Coogan and Arcak, 2016, Lovisari et al., 2014, Nilsson and Como, 2020), etc. Meanwhile, the relevant studies that consider both network resilience and traffic flow networks involve distributed routing, local routing (Como et al., 2014, Como et al., 2011), etc. Unlike the various studies mentioned above, this paper considers both traffic flow network and resilience. It constructs an equilibrium selection problem of traffic flow, in which optimal traffic flow can maintain the complete transmission flow under specific link capacity reduction. Secondly, we propose a distributed discrete algorithm to solve the proposed equilibrium selection problem of traffic flow over a time-varying undirected communication network. It is more realistic than the research of communication through a time-invariant communication network (Liu et al., 2017, Shi et al., 2015, Xu et al., 2015) because the sensor information interaction installed in the traffic network cannot guarantee real-time and synchronization. Thus, the algorithm has broad application prospects. Beyond these, compared with the existing research Lü et al., 2018, Nedić and Olshevsky, 2016, Nedic et al., 2017, Shi et al., 2015, Xu et al., 2015, the equality constraint in the optimization problem studied in this paper are affine rather than unconstrained or simple equality constraints in resource allocation. On this basis, we analyze the convergence of the proposed algorithm, and the theoretical analysis shows that the proposed algorithm can achieve a geometric convergence rate under time-varying communication among agents. For the simulation section, the results show that the convergence rate of the proposed algorithm is related to the connectivity of the time-varying undirected graph . In addition, when the graph is 2-Connected or 3-Connected, the algorithm’s convergence rate is faster than the time-invariant graph.
This paper is organized as follows. Some preliminary notations, graph theory, and incidence matrix are presented in Section 2. Section 3 focuses on the formulation of the studied problem and its corresponding analysis, including necessary definitions and assumptions for the subsequent studies. For the proposed problem described in Section 3, the designed algorithm is then presented in Section 4. The convergence of the proposed algorithm is analyzed in Section 5, followed by the descriptions of a numerical example and the simulation results in Section 6.
Section snippets
Preliminaries
In this subsection, we review some basic concepts of algebraic graph theory and incidence matrix. Before proceeding, we first define some preliminary notations to be used throughout the paper. Denote as the set of real numbers and as the set of nonnegative real numbers. Denote as the set of positive integer. Let be finite set, be the cardinality of . Denote 1 and 0 as the column vectors with all entries equal to one and zero, respectively. For any matrix , let
Problem formulation and analysis
In this subsection, we give some necessary definitions, then we propose the studied problem using resilience constraints and traffic flow network framework (Como et al., 2012). The target of the studied problem is to find the optimal traffic flow in the feasible set with strongly convex separable general cost functions over links. For the optimal equilibrium flow obtained by the proposed algorithm, intelligent road pricing (Sharon, 2021) and manual traffic control (Parr & Wolshon, 2016) are
Algorithm design
In this subsection, we list several typical research on algorithms related to the problem studied in this paper, which states in the above Table 1 . It shows that the communication graphs among agents in the current studies can be divided into two categories, i.e., time-invariant undirected graph and time-varying directed graph. In the following, we will start to analyze our problem and then propose our algorithm. Denote as the Lagrangian multiplier associated with equality constraint
Supporting lemmas and definitions
Before analyzing the convergence of the proposed algorithm, we introduce some additional symbols and lemmas, which will be utilized in the subsequent analysis.
Theorem 1 Suppose that are sequences such that for all positive integers and for each , we have an arrow , that is, where the constants(gains) are nonnegative and satisfy . Then Small Gain Theorem Nedic et al., 2017
Since involving a cyclic structure
Simulation
In this subsection, we present a numerical example to illustrate the effectiveness of the proposed algorithm in the multi-agent simulation environment. In Fig. 2, the numbered circles and the numbered edges represent the intersections and directed links of the traffic flow network, respectively. We set the constant external inflow of the flow network, the fuel cost per mile and the minimum node residual capacity over the flow network as , 0.1, and , respectively. The step size of our
Conclusion
This paper studies the traffic network with one source and destination and assumes that the inflow of the source node is time-invariant. Combined with the strong resilience in the traffic flow network, we propose an optimal selection problem of equilibrium flow and design a distributed optimization algorithm to tackle the problem. The proposed algorithm allows the communication network among agents to be time-varying undirected, which has better prospects than time-invariant graphs. As
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work was supported by the Analytical Center for the Government of the Russian Federation (IGK 000000D730321P5Q0002), agreement No. 70-2021-00141.