User Equilibrium Traffic Assignment Research Articles

Formulating multi-class user equilibrium using mixed-integer linear programming

We introduce an approach to formulate and solve the multi-class user equilibrium traffic assignment as a mixed-integer linear programming (MILP) problem. Compared to simulation approaches, the analytical MILP formulation makes the solution of network assignment problems more tractable. When applied in a multi-class context, it obviates the need to assume a symmetrical influence between classes and thereby allows richer traffic behavior to be taken into account. Also, it integrates naturally in optimization problems such as maintenance planning and traffic management. We develop the model and apply it for the Sioux Falls network, showing that it outperforms the traditional Beckmann-based and MSA approaches in smaller-scale problems. Further research opportunities lie in developing extensions of MILP-based assignment, with different variants of user equilibrium or dynamic assignment, and in improving the model and solution algorithms to allow large-scale application. • We formulate multi-class static traffic assignment using MILP. • The new formulation is analytical and therefore prescriptive. • We compare the performance of MILP with other existing methods in the Sioux Falls network. • This formulation opens a new axis for future works.

Development of a solution algorithm to solve the user equilibrium traffic assignment problem with soft link capacity constraints

Development of a solution algorithm to solve the user equilibrium traffic assignment problem with soft link capacity constraints

Multi-objective Eco-Routing Model Development and Evaluation for Battery Electric Vehicles

This paper develops a multi-objective eco-routing algorithm (eco- and travel time-optimum routing) for battery electric vehicles (BEVs) and internal combustion engine vehicles (ICEVs) and investigates the network-wide impacts of the proposed multi-objective Nash optimum (user equilibrium) traffic assignment on a large-scale network. Unlike ICEVs, BEVs are more energy efficient on low-speed arterial trips compared with highway trips. Different energy consumption patterns require different eco-routing strategies for ICEVs and BEVs. This study found that single-objective eco-routing could significantly reduce the energy consumption of BEVs but also significantly increase their average travel time. Consequently, the study developed a multi-objective routing model (eco- and travel time-routing) to improve both energy and travel time measures. The model introduced a link cost function that uses the specification of the value of time and the cost of fuel/energy. The simulation study found that multi-objective routing could reduce BEV energy consumption by 13.5%, 14.2%, 12.9%, and 10.7%, as well as ICEV fuel consumption by 0.1%, 4.3%, 3.4%, and 10.6% for “not congested, “slightly congested,”“moderately congested,” and “highly congested” conditions, respectively. The study also found that multi-objective user equilibrium routing reduced the average vehicle travel time by up to 10.1% compared with the standard user equilibrium traffic assignment for highly congested conditions, producing a solution closer to the system optimum traffic assignment. The results indicate that the proposed multi-objective eco-routing strategy can reduce vehicle fuel/energy consumption effectively with minimum impacts on travel times for both BEVs and ICEVs.

Freeway Network Travel Time Reliability Analysis Methodology and Software Tool Development

Applications in the field often require analysis of freeway travel time reliability (TTR) at the network level. While micro-simulation is suitable for performing network-level analysis, the computational burden can become unreasonable when TTR analysis is factored in. As for macro-simulation, most network analysis studies often use simplified link performance functions to represent the travel time and flow relationship. Such functions are not generally sensitive to the range of geometric and traffic conditions that can influence freeway facility operations. This research extends the Highway Capacity Manual (HCM) freeway TTR analysis methodology to the network level. The proposed freeway network TTR analysis methodology generates scenarios that represent the impacts of origin–destination (OD) demand variations, weather events, incident events, and work zone events on freeway network travel time. For each scenario, the methodology performs user equilibrium (UE) traffic assignment and the HCM freeway facility core methodology is applied to represent the travel time and flow relationship. The method of successive average approach is applied to solve the UE traffic assignment. Finally, scenario travel times (and/or other performance measures) are aggregated into various distributions of interest, such as the network-, facility-, and OD-level distributions, and TTR performance measures are calculated at the three different levels. A software tool is developed using C# language on the .NET Framework. The software tool provides a convenient and efficient approach for transportation planners and researchers to conduct the freeway network TTR analysis, which helps to bridge the gap between research and practice.

An improved parallel block coordinate descent method for the distributed computing of traffic assignment problem

An improved parallel block coordinate descent (iPBCD) algorithm for solving the user equilibrium traffic assignment problem is proposed. The iPBCD algorithm is developed based on the parallel block coordinate descent algorithm (PBCD). The hybrid flow update policy is investigated to enhance the robustness and performance of the PBCD algorithm. Two update order rules, namely a cyclic rule and a greedy rule are compared for block indices. Then, the block size is optimized using a sensitivity analysis test. Finally, five index-grouping rules are tested for comparative purposes. Numerical experiments indicate that index-grouping rules have a significant influence on convergence: the information-based drop-out rule performs better in terms of convergence and efficiency.

Optimal Planning of Charging Stations in Coupled Transportation and Power Networks Based on User Equilibrium Conditions

In this article, we propose a new bi-level approach for the location and sizing of charging stations (CSs) based on the integration of power distribution and traffic networks. At the lower level, the user choices and the energy demands are modeled through the user equilibrium (UE) traffic assignment conditions that are derived for the case of electric vehicles (EVs). At the higher level, an optimization problem is formalized for the optimal planning of locations, sizes, and unit prices of a set of new charging stations in a territory, which embeds the UE conditions and the power flow model as constraints. The main novelties lie in the integration of traffic and electric networks, the optimization problem formalization, and the inclusion of detailed electrical issues [such as the complete alternating current (AC) power flow equations and the modeling of CSs with different phases]. The developed model has been applied to a case study in the Genoa Municipality. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In the past decade, particular attention has been posed on EVs and renewable resources to reduce pollutants’ emissions. Critical issues regard the setting of adequate recharging infrastructure and the integration of huge distributed and intermittent loads and production into the electrical grid. This article aims to provide a bi-level optimization model that can be used by companies or public administrations that want to install charging stations in a territory. In particular, the transportation and the electrical networks are integrated: at a lower level, the UE traffic assignment conditions are derived to represent users’ choices and to assess the energy demand. At the upper level, an optimization problem is formalized, to minimize costs of installation and power losses and to maximize benefits, which embeds as constraints the UE conditions for the case of EVs and the model of the electrical grid (load flow equations, bounds over-voltage, apparent power, etc.).

A computationally efficient metric for identification of critical links in large transportation networks

Maintaining safety and resilience in transportation system requires identification of critical components. The existing transportation link criticality metrics typically require multiple traffic assignments to perform a full scan of the network. Link removals create computational burden and can cause network disconnectivity which makes it problematic to run traffic assignment. The authors previously introduced Link Criticality Index (LCI) that identifies the criticality ranking within a single User Equilibrium (UE) traffic assignment using Frank-Wolfe (FW) algorithm. While LCI is shown to provide balanced rankings with respect to connectivity and flow conditions, its computational efficiency diminishes for larger networks due to the need for path enumeration. This paper formulates an adjusted LCI to make it computationally efficient for larger networks. Adjusted LCI utilizes 1) viable routes instead of the complete list of enumerated paths; and, 2) a path-based traffic assignment algorithm (Gradient Projection) to replace FW. The consistency of rankings between original/adjusted LCIs, and other measures from the literature are analyzed and compared for three networks, and through an experimental setup. The proposed method is also applied on a large-scale transportation network, i.e., Chicago network. The results show that both modifications are effective and yield faster results without compromising LCI's advantages.

Application of Traffic Weighted Multi-Map Optimization Strategies to Traffic Assignment

Traffic Assignment Problem (TAP) is a critical issue for transportation and mobility models that deals mainly with the calculus and delivery of best-cost routes for the trips in a traffic network. It is a computationally complex problem focused on finding user equilibrium (UE) and system optimum (SO). The Traffic Weighted Multi-Maps (TWM) technique offers a new perspective for TAP calculus, based on routing decisions using different traffic network views. These TWM are complementary cost maps that combine physical traffic networks, traffic occupation data, and routing policies. This paper shows how evolutionary algorithms can find optimal cost maps that solve TAP from the SO perspective, minimizing total travel time and providing the best-cost routes to vehicles. Several strategies are compared: a baseline algorithm that optimizes the whole network and two algorithms based on extended k-shortest path mappings. Algorithms are analyzed following a simulation-optimization methodology over synthetic and real traffic networks. Obtained results show that TWM algorithms generate solutions close to the static UE traffic assignment methods at a reasonable computational cost. A crucial aspect of TWM is its good performance in terms of optimal routing at the system level, avoiding the need for continuous route calculus based on traffic status data streaming.

Strategic Behavior of In-Motion Wireless Charging Aggregators in the Electricity and Transportation Networks

This paper presents a framework to determine the strategic bidding of the wireless charging aggregators that operate wireless charging stations to provide in-motion charging services for the electric vehicles. The proposed framework captures the interactions between the electricity and transportation networks through competition among the wireless charging aggregators. Wireless charging aggregators participate in the wholesale electricity market to minimize the cost of gained energy from the bulk electricity network. In the retail wireless charging market, wireless charging aggregators compete to maximize their revenue by offering energy to the electric vehicles that are traveling on the transportation links. The strategic behavior of the wireless charging aggregators highlights the interdependence between the electricity and transportation infrastructure systems. In each system, the strategic behavior of the entities is determined by formulating a bi-level optimization problem which is further transformed into a mathematical problem with equilibrium constraints. The upper-level problem maximizes the profit of the entities while the lower level problem ensures the balance between the demand and supply in each infrastructure system. The demand and supply balance is guaranteed by the optimal power flow in the electricity network and the user equilibrium traffic assignment in the transportation network.

A parallel computing approach to solve traffic assignment using path-based gradient projection algorithm

This paper presents a Parallel Block-Coordinate Descent (PBCD) algorithm for solving the user equilibrium traffic assignment problem. Most of the existing algorithms for the user equilibrium-based traffic assignment problem are developed and implemented sequentially. This paper aims to study and investigate the parallel computing approach to utilize the widely available parallel computing resources. The PBCD algorithm is developed based on the state-of-the-art path-based algorithm, i.e., the improved path-based gradient projection algorithm (iGP). The computationally expensive components in the iGP are identified and parallelized. A parallel block-coordinate method is proposed to replace the widely used Gauss-Seidel method for the procedure of path flow adjustment. A new rule is proposed to group OD pairs into different blocks. The numerical examples show that the implemented PBCD algorithm can significantly reduce the computing time.

Integrated distribution expansion planning considering stochastic renewable energy resources and electric vehicles

Renewable energy resources and transport electrification have become essential components of modern and future distribution planning. Despite the impetus of economic and environmental benefits, the uncertainty brought in poses challenges. In this paper, we propose an integrated expansion planning framework based on a multiobjective mixed-integer nonlinear program. The aim is to minimize the net present value of investments considering feeder routing, substation alterations and construction while maximizing the utilization of proposed charging stations. Distributed generation and load uncertainties, recast in two-stage stochastic programming, are tackled with a scenario generation and reduction technique using a probabilistic approach with Kantorovich metrics. The final number of the scenarios are validated with an alternative clustering method. A flow-based location–allocation theory with user equilibrium traffic assignment model is exploited to site charging stations. The sizing problem is determined using continuous-time Markov chain modeling. The proposed framework is solved with a multiobjective Tchebycheff decomposition-based evolutionary algorithm and tested on a modified 54 bus distribution network and 25 transportation node system. Numerical results demonstrate the capability of the proposed method. Distribution planning authorities can benefit from the presented approach to make intertemporal investment decisions while maintaining the quality of system performance.

A Two-Class Stochastic Network Equilibrium Model under Adverse Weather Conditions

Adverse weather condition is one of the inducements that lead to supply uncertainty of an urban transportation system, while travelers’ multiple route choice criteria are the nonignorable reason resulting in demand uncertainty. This paper proposes a novel stochastic traffic network equilibrium model considering impacts of adverse weather conditions on roadway capacity and route choice criteria of two-class mixed roadway travellers on demand modes, in which the two-class route choice criteria root in travelers’ different network information levels (NILs). The actual route travel time (ARTT) and perceived route travel time (PRTT) are considered as the route choice criteria of travelers with perfect information (TPI) and travelers with bounded information (TBI) under adverse weather conditions, respectively. We then formulate the user equilibrium (UE) traffic assignment model in a variational inequality problem and propose a solution algorithm. Numerical examples including a small triangle network and the Sioux Falls network are presented to testify the validity of the model and to clarify the inner mechanism of the two-class UE model under adverse weather conditions. Managerial implications and applications are also proposed based on our findings to improve the operation efficiency of urban roadway network under adverse weather conditions.

Estimation of the value of travel time and of travel time reliability for heterogeneous drivers in a road network

Travel time reliability has been recognized as an important factor in cost–benefit analysis in a transportation network. To estimate the benefit and cost of travel time reliability, several network models have been proposed to estimate the monetary values both of travel time and of travel time reliability based on the driver’s route choice behavior. In this study, we propose a network model that addresses monetary values that are specific to each origin–destination pair in a road network. The proposed model is formulated as a utility maximization problem with constraints in a manner similar to studies that address the value of travel time only. The proposed model has the same equilibrium conditions as a multiclass user equilibrium traffic assignment problem with elastic demand in which the risk-averse driver’s route choice behavior is explicitly considered. By accounting for data availability, we propose a parameter estimation method based on the maximum likelihood method with two alternative approaches; namely link flow-based and route flow-based approaches. However, the two proposed methods require link flow data only. Numerical experiments are carried out to demonstrate the performance of the two proposed methods together with some insightful findings.

Optimal traffic management policies for mixed human and automated traffic flows

Abstract Although Autonomous Vehicles (AVs) will enhance mobility and safety, their impact on congestion is not clear yet. AVs may increase roadway capacity due to their connectivity features. The capacity enhancement highly depends on the AV proportion in traffic. This study models user equilibrium traffic assignment when the link capacity is a function of AV proportion of traffic. The mixed traffic flow of AVs and human-driven vehicles is considered as a multiclass traffic assignment problem. This problem is formulated as a non-linear complementarity problem which is solved to find optimal traffic management policies. We show that simple policies such as AV exclusive links can improve network performance in mixed traffic of AVs and human-driven vehicles. We also show that if these policies are implemented the network performance would be very close to system optimal condition even when users choose their routes selfishly following a user equilibrium. Results of numerical examples for a real size network show that management policies can decrease the gap between user equilibrium and system optimal to less than 1%.

Evaluation of navigation based on system optimal traffic assignment for connected cars

Recently, many cars have become connected to internet. In the near future, almost all cars will be connected cars. Such a connected car will automatically drive according to a navigation system. Conventional car navigation systems are based on User Equilibrium (UE) traffic assignment. However, System Optimal (SO) traffic assignment is better than UE traffic assignment. To realise SO traffic assignment, complete traffic information is required. When connected cars become ubiquitous, all traffic information will be gathered into the cloud. Therefore, a cloud-based navigation system can provide SO-based navigation to connected cars. An SO-based navigation method in which the cloud collects traffic information from connected cars, computes SO traffic assignments, and recommends SO routes to users was recently proposed. In this paper, we evaluate this SO-based navigation method in detail.

Modeling joint activity-travel patterns in pedestrian networks with use of Wi-Fi data

Abstract This paper proposes a new activity-based equilibrium model to solve a joint activity-travel pattern scheduling problem in pedestrian networks. A novel supernetwork platform, pedestrian-joint-activity-time-space, is proposed to capture the complex structure of pedestrian networks and intragroup interactions on joint activity choices of pedestrians. The joint activity-travel pattern scheduling problem is transformed into an extended static user equilibrium traffic assignment model with side constraints for group interactions. A heuristic solution method is proposed to solve the user equilibrium problem concerned. A simplified pedestrian network is used to illustrate the convergence of the proposed heuristic solution method. Furthermore, the use of Wi-Fi tracking data from mobile phones to generate the activity patterns of pedestrians and their interactions is also investigated with a real case study in a selected area of The Hong Kong Polytechnic University.

Optimal planning of charging stations and electric vehicles traffic assignment: a bi-level approach

A new bi-level approach is proposed for the location and sizing of charging stations, considering both the transportation and energy demands. The lower level considers the User Equilibrium traffic assignment conditions for Electric Vehicles (EVs) which are derived and inserted as constraints in the overall optimization problem. The higher level presents the formalization of an optimization problem for the optimal planning of locations, sizes and unit prices of a set of new charging stations in a territory characterized by the presence of an already existing set of charging stations. A case study in the Genoa Municipality is considered for the application of the proposed model.

Reduced gradient algorithm for user equilibrium traffic assignment problem

ABSTRACT A path-based algorithm is developed for the static traffic assignment problem (TAP). In each iteration, it decomposes the problem into origin-destination (OD) pairs and solves each subproblem separately using the Wolfe reduced gradient (RG) method. This method reduces the dimensions of each single-OD subproblem by selecting a basic path between the OD pair and reformulating the subproblem in terms of the nonbasic paths. A column generation technique is included to avoid path enumeration in large scale networks. Also, some speed-up techniques are designed to improve the computational efficiency. The algorithm shifts flows from costlier paths to cheaper paths; however, the amount of flow shifted from a costlier path is proportional to not only the travel time but also the flow on the path. It is applied to the Philadelphia and Chicago test problems, while different strategies for choosing the basic paths are examined. The RG algorithm shows an excellent convergence to relative gaps of the order of 1.0E-14 when compared against several reference TAP algorithms.

Network performance evaluation under disruptive events through a progressive traffic assignment model

The purpose of this paper is to present an assignment model as a basis for evaluating the performance of a traffic network, capable of describing its evolution immediately after the occurrence of a disruptive event. First of all, a User-Equilibrium traffic assignment problem is solved in order to obtain an estimation of the system state before the disruption. Starting from the critical event, a Progressive Assignment procedure is performed in order to obtain reasonable traffic assignments on the network, taking into account the users’ tolerance to increases in travel times as well as the inherent inertia of the system. Therefore a metric for the description of the network performance is proposed as well as implementation of the model on a test network.

Reservoir-based surrogate modeling of dynamic user equilibrium

This paper proposes a new dynamic user equilibrium (DUE) traffic assignment model using reservoir-based network reduction techniques and surrogate dynamic network loading models. A traffic network is decomposed into a reservoir structure, and the DUE problem is formulated as a variational inequality, with an embedded surrogate model for the path delay operator to describe traffic dynamics at the reservoir level. The surrogate model is further enhanced by the reproducing kernel Hilbert space and adaptive sampling to reduce approximation error and improve computational efficiency. To solve the proposed surrogate-based DUE problem on reduced networks, we develop a customized algorithm that integrates the kernel trick with the generalized projection framework. A pre-computation scheme is proposed, which calculates and stores the of kernel matrices and vectors, could further reduce the computational burden. Numerical experiments of the proposed methods show significant reduction of the computational times, by up to 90%, while maintaining low approximation errors (MAPE below 6%), when compared to the exact models and solution methods.