Stochastic User Equilibrium Research Articles

Congestion patterns of electric vehicles with limited battery capacity.

The path choice behavior of battery electric vehicle (BEV) drivers is influenced by the lack of public charging stations, limited battery capacity, range anxiety and long battery charging time. This paper investigates the congestion/flow pattern captured by stochastic user equilibrium (SUE) traffic assignment problem in transportation networks with BEVs, where the BEV paths are restricted by their battery capacities. The BEV energy consumption is assumed to be a linear function of path length and path travel time, which addresses both path distance limit problem and road congestion effect. A mathematical programming model is proposed for the path-based SUE traffic assignment where the path cost is the sum of the corresponding link costs and a path specific out-of-energy penalty. We then apply the convergent Lagrangian dual method to transform the original problem into a concave maximization problem and develop a customized gradient projection algorithm to solve it. A column generation procedure is incorporated to generate the path set. Finally, two numerical examples are presented to demonstrate the applicability of the proposed model and the solution algorithm.

Passenger flow control with multi-station coordination in subway networks: algorithm development and real-world case study

ABSTRACTThis paper addresses the problem of passenger flow control in a multi-line subway network during peak hours. A new multi-station coordinated passenger flow control model is proposed to simultaneously adjust the number of inbound and transfer passengers entering multiple stations or lines. The implementation of passenger flow control governs the redundant passengers to queue at given facilities, which is similar to ramp metering that regulates the number of vehicles entering a highway segment. The proposed model is a bi-level programming, whose upper level aims to optimize the system performance with different passenger flow control strategies, while the lower level is a logit-based stochastic user equilibrium assignment problem under a given strategy, considering passenger flow evolution, dynamic path cost, and route choice. An algorithm, integrating the method of successive averages with genetic algorithm, is developed to solve the model. A real-world case study is conducted to examine the validity.

Congestion tolling - dollars versus tokens: A comparative analysis

The case for some form of congestion tolling has long been made given the extent of traffic congestion in most urban transportation networks. However, there is little consensus on whether this tolling should be in the form of dollars (traditional congestion pricing schemes or price regulation) or in the form of tokens (credit based mobility schemes or quantity regulation). Although the comparison of price and quantity regulation has received significant attention in the economics community, the literature is relatively sparse in the context of transportation systems. The two systems differ not only in terms of transaction costs but also in their efficiency in dealing with demand and capacity shocks, frequent in road transport systems. This difference has been explored by economists in market contexts but the possible parallel in transport systems has largely been overlooked. This paper develops a methodology to compare the performance of price and quantity instruments under uncertainty using a simple transportation network consisting of parallel highway routes and a public transport alternative. Tolls (in dollars or tokens) are differentiated across roads and the permits or tokens can be traded across roads at fixed rates. The demand for each route is determined by a logit mixture model and the supply consists of static congestion. The comparison is based on the optimum social welfare which is computed for each instrument by solving a non-convex optimization problem involving stochastic user equilibrium constraints. Equity considerations are also examined.Numerical experiments across a wide range of demand/supply inputs lead us to the following insights. First, when the tolls in dollars and tokens can be fully adapted to demand and supply shocks (i.e. are day-to-day adaptive), both instruments are equivalent and attain the same optimum social welfare. Second, when the tolling system is not day-to-day adaptive and the supply of tokens is fixed, the quantity instrument performs better than the price instrument typically when the marginal external congestion function has a steeper slope than the demand function. Third, when the tolling system is non-adaptive, but the supply of tokens is day-to-day adaptive, the quantity instrument always outperforms the price instrument in terms of social welfare. Fourth, when the toll revenues and permits are distributed uniformly, this has a positive equity effect but the quantity instrument is more equitable than the price instrument. In view of these results, we argue the case for quantity control in the presence of uncertainty and strongly convex congestion costs.

Bi-Level Programming Model for Dynamic Reversible Lane Assignment

To improve the space utilization and traffic capacity of a signalized junction, this paper proposes a dynamic reversible lane assignment method for approaches of signalized junctions that consider the game equilibrium between road users and traffic controllers. To theoretically analyze the behaviors of the players involved in the leader-follower strategic game, a bi-level programming model is established. To minimize the total queue length of approaches at signalized junctions, the upper model dynamically optimizes the reversible lane assignments and can be solved with the enumeration method or the Monte Carlo algorithm. The lower model outputs the traffic assignment at road sections using a Logit-based stochastic user equilibrium model that is solved by the method of successive averages. The general impedance of road section in lower model consists of the road travel time cost and the intra-time cost at the signalized junction connected with current road section. In addition, the interaction between the two level is simulated in an iterative optimization procedure. Finally, this paper uses two numerical experiments to validate the proposed approach.

An <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math> </inline-formula>-Path Logit-Based Stochastic User Equilibrium Model

This paper proposes a new traffic assignment model, named an $N$ -path logit-based stochastic user equilibrium (NPSUE), which considers not only road users’ perceptual errors in path choices, but also users’ heterogeneity that allows the numbers of the limited path sets for each user to be different. The property of the $N$ -path user equilibrium (NPUE) proposed in Dung-Ying Lin (2014) is first reanalyzed, and then the optimization program for the NPSUE models as well as the property of the optimal solution are explicitly discussed. For each type of users, the NPSUE assignment results are consistent with the logit-assignment results in their limited path set. Both the NPUE and NPSUE models are tested in three road networks: an simple grid network, Nguyen and Dupuis’ network and Sioux Falls network. The numerical results show that (1) the NPUE can be equivalent to the UE under certain conditions, although the users don’t have full information on the available paths; (2) the proposed NPSUE may provide an ideal equilibrium state for both system managers and road users; (3) the provision of appropriate path guidance information is beneficial to both system managers and road users; otherwise, the provision of excessive information may decrease the performance of the road network, and (4) under the NPSUE conditions, the provision of excessive information may decrease users’ satisfaction.

The health impacts of weekday traffic: A health risk assessment of PM2.5 emissions during congested periods

Little work has accounted for congestion, using data that reflects driving patterns, traffic volume, and speed, to examine the association between traffic emissions and human health. In this study, we performed a health risk assessment of PM2.5 emissions during congestion periods in the Greater Toronto and Hamilton Area (GTHA), Canada. Specifically, we used a micro-level approach that combines the Stochastic User Equilibrium Traffic Assignment Algorithm with a MOVES emission model to estimate emissions considering congestion conditions. Subsequently, we applied a concentration-response function to estimate PM2.5-related mortality, and the associated health costs. Our results suggest that traffic congestion has a substantial impact on human health and the economy in the GTHA, especially at the most congested period (7:00am). Considering daily mortality, our results showed an impact of 206 (boundary test 95%: 116; 297) and 119 (boundary test 95%: 67; 171) deaths per year (all-cause and cardiovascular mortality, respectively). The economic impact from daily mortality is approximately $1.3 billion (boundary test 95%: 0.8; 1.9), and $778 million (boundary test 95%: 478; 981), for all-cause and cardiovascular mortality, respectively. Our study can guide reliable projections of transportation and air pollution levels, improving the capability of the medical community to prepare for future trends.

Maximizing Network Throughput under Stochastic User Equilibrium with Elastic Demand

Most of the existing studies adopt the fixed-demand equilibrium formulation to model drivers’ route choice when studying network throughput maximization problem. Travelers’ reactions to the increased origin-destination (O-D) travel cost and network congestion level are less considered in the problem. Note that travelers can cancel the trip or use other modes to travel if the road network is congested. This study aims to address this gap by analyzing the maximum network throughput problem using the formulation of Logit-based SUE with elastic demand (SUEED). The Logit-based SUEED problem not only models the drivers’ route choice according to the SUE principle, but also estimates the equilibrium O-D demand by factoring the effect of expected perceived O-D travel time on O-D demand. A bi-level programming problem is proposed to characterize the maximum network throughput based on the Logit-based SUEED problem. The sensitivity analysis for the Logit-based SUEED problem is presented and incorporated into the solution algorithm for the proposed problem. A numerical example demonstrates the effectiveness of the proposed sensitivity-based solution algorithm. This study finds that under the SUEED condition, the maximum network throughput decreases monotonically when travelers’ knowledge level of traffic conditions increases (less travel time perception error). It implies that promoting advanced traveler information system ATIS may not serve to foster more number of trips by travelers and make more use of physical network capacity.

Optimizing a desirable fare structure for a bus-subway corridor.

This paper aims to optimize a desirable fare structure for the public transit service along a bus-subway corridor with the consideration of those factors related to equity in trip, including travel distance and comfort level. The travel distance factor is represented by the distance-based fare strategy, which is an existing differential strategy. The comfort level one is considered in the area-based fare strategy which is a new differential strategy defined in this paper. Both factors are referred to by the combined fare strategy which is composed of distance-based and area-based fare strategies. The flat fare strategy is applied to determine a reference level of social welfare and obtain the general passenger flow along transit lines, which is used to divide areas or zones along the corridor. This problem is formulated as a bi-level program, of which the upper level maximizes the social welfare and the lower level capturing traveler choice behavior is a variable-demand stochastic user equilibrium assignment model. A genetic algorithm is applied to solve the bi-level program while the method of successive averages is adopted to solve the lower-level model. A series of numerical experiments are carried out to illustrate the performance of the models and solution methods. Numerical results indicate that all three differential fare strategies play a better role in enhancing the social welfare than the flat fare strategy and that the fare structure under the combined fare strategy generates the highest social welfare and the largest resulting passenger demand, which implies that the more equity factors a differential fare strategy involves the more desirable fare structure the strategy has.

Optimal toll of new highway in the equilibrium framework of heterogeneous households' residential location choice

Because of limited budget and the consideration of transportation demand management, most of the new highways will be tolled after putting them into use. Meanwhile, lots of people belonging to different sectors of society flocked to city to search for jobs and better living each year. Therefore, for the government departments, one of most important issues is to determine the optimal toll of new highway under the background of ever-increasing households. This paper proposes an optimal toll model in the equilibrium framework of heterogeneous households' residential location choice. It is formulated as a bi-level program. In the upper-level subprogram, government departments determine an optimal toll of new highway in order to minimize urban system’s total travel time. However, the construction and toll charge of the new highway will totally change urban traffic condition, and thus under the assumption of a complete market economy it will further affect the new households' location and commute behavior. Here, new households' location choice behavior has been modeled based on reference-dependent theory and household balance model. And their route choice behavior has been formulated as the multi-class multi-criteria stochastic user equilibrium model. This combined residential location choice and traffic assignment problem has been formulated by fixed point theory as the lower-level program. Finally, a genetic algorithm based solution approach is used to solve this optimization problem. In addition, a set of numerical experiments has been conducted to demonstrate the properties of this problem and the performance of our proposed model. And some interesting conclusions have been given.

A select link analysis method based on logit\u2013weibit hybrid model

Select link analysis provides information of where traffic comes from and goes to at selected links. This disaggregate information has wide applications in practice. The state-of-the-art planning software packages often adopt the user equilibrium (UE) model for select link analysis. However, empirical studies have repeatedly revealed that the stochastic user equilibrium model more accurately predicts observed mean and variance of choices than the UE model. This paper proposes an alternative select link analysis method by making use of the recently developed logit–weibit hybrid model, to alleviate the drawbacks of both logit and weibit models while keeping a closed-form route choice probability expression. To enhance the applicability in large-scale networks, Bell’s stochastic loading method originally developed for logit model is adapted to the hybrid model. The features of the proposed method are twofold: (1) unique O–D-specific link flow pattern and more plausible behavioral realism attributed to the hybrid route choice model and (2) applicability in large-scale networks due to the link-based stochastic loading method. An illustrative network example and a case study in a large-scale network are conducted to demonstrate the efficiency and effectiveness of the proposed select link analysis method as well as applications of O–D-specific link flow information. A visualization method is also proposed to enhance the understanding of O–D-specific link flow originally in the form of a matrix.

Selfish routing equilibrium in stochastic traffic network: A probability-dominant description.

This paper suggests a probability-dominant user equilibrium (PdUE) model to describe the selfish routing equilibrium in a stochastic traffic network. At PdUE, travel demands are only assigned to the most dominant routes in the same origin-destination pair. A probability-dominant rerouting dynamic model is proposed to explain the behavioral mechanism of PdUE. To facilitate applications, the logit formula of PdUE is developed, of which a well-designed route set is not indispensable and the equivalent varitional inequality formation is simple. Two routing strategies, i.e., the probability-dominant strategy (PDS) and the dominant probability strategy (DPS), are discussed through a hypothetical experiment. It is found that, whether out of insurance or striving for perfection, PDS is a better choice than DPS. For more general cases, the conducted numerical tests lead to the same conclusion. These imply that PdUE (rather than the conventional stochastic user equilibrium) is a desirable selfish routing equilibrium for a stochastic network, given that the probability distributions of travel time are available to travelers.

TRAFFIC ASSIGNMENT MODEL OF AL-AMARAH CITY

The initiation of this study was made with the objective of loading or assigning the flow of traffic in the base and target year (expressed as a matrix identifying the number of trips generated and distributed in the previous study from each Traffic analysis zone (TAZ) to every other within the study area) on a road network of Al-Amarah City. The traffic assignment process is usually implemented in two steps. The first step is constructing the network dataset of road network of Al-Amarah city for obtaining the optimal route between each node to all other nodes depending on two criteria; distance and time for evaluation the accessibility of all nodes in the road network. The second step uses stochastic user equilibrium and system optimum assignment models to assign the traffic distributed between all possible origins and destinations to the road network itself. The analysis is done using TransCAD software ver. 4.5 and Network Analysis/Arc GIS 10.3 for Al-Amarah City. Results showed that in the base year, The results of the stochastic user equilibrium assignment model there is no link in Al-Amarah network has v/c ratio between the range (0.81-1.00) and over which represent level of service equal to or more than D. The overall evaluation of Al-Amarah road network is level of service (LOS) equal to B and there is an improvement in the work of Al-Amarah road network by using the system optimum model. In the target year, the results of two assignment models are proximally similar. The black color (v/c >1.0) is appeared clearly in the assignment result map especially on the two main bridges in the city (Al-Yugoslavi Bridge and Al-Jumhuriah Bridge). When the red color (LOS E) is appeared in the links in near the centers of sectors (1,2, and 4). The most links ranges between (LOS D and LOS C) with v/c ratio Range (0.70-0.90).

A day-to-day route flow evolution process towards the mixed equilibria

This study investigates a travelers’ day-to-day route flow evolution process under a predefined market penetration of advanced traveler information system (ATIS). It is assumed that some travelers equipped with ATIS will follow the deterministic user equilibrium route choice behavior due to the complete traffic information provided by ATIS, while the other travelers unequipped with ATIS will follow the stochastic user equilibrium route choice behavior. The interaction between these two groups of travelers will result in a mixed equilibrium state. We first propose a discrete day-to-day route flow adjustment process for this mixed equilibrium behavior by specifying the travelers’ route adjustment principle and adjustment ratio. The convergence of the proposed day-to-day flow dynamic model to the mixed equilibrium state is then rigorously demonstrated under certain assumptions upon route adjustment principle and adjustment ratio. In addition, without affecting the convergence of the proposed day-to-day flow dynamic model, the assumption concerning the adjustment ratio is further relaxed, thus making the proposed model more appealing in practice. Finally, numerical experiments are conducted to illustrate and evaluate the performance of the proposed day-to-day flow dynamic model.

Direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem

ABSTRACTThis paper proposes simple and direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem. It is only necessary to account for random variables independent of link flows by performing a simple transformation of the perceived link travel time with a normal distribution. At every iteration of a Monte-Carlo simulation procedure, the values of the random variables are sampled based on their probability distributions, and then a regular deterministic user equilibrium assignment is carried out to produce link flows. The link flows produced at each iteration of the Monte-Carlo simulation are averaged to yield the final flow pattern. Two test networks demonstrate that the proposed algorithms and the traditional algorithm (the Method of Successive Averages) produce similar results and that the proposed algorithms can be extended to the computation of the case in which the random error term depends on measured travel time.

On the Equivalence Between SUE and Fixed-Point States of Day-to-Day Assignment Processes with Serially-Correlated Route Choice

We propose a new theoretical framework of deterministic processes of traffic assignment able to include day-to-day correlation of the random terms. According to the prevailing interpretation of random utility models, random terms are regarded as individual specific. Correlation is justified by persistence of habits and unobservables. The framework includes the deterministic sequence of systematic utilities based on a learning filter, the stochastic process of the random terms based on a stationary autoregressive structure with i.i.d. Gumbel or multivariate normal one-day marginals, and the resulting stochastic process of choice which generally is not Markov. The fixed point states of the choice process are the classical logit and probit Stochastic User Equilibrium (SUE). The linkage between flows at any day and transition flows is made explicit, and, by this, a new perspective on the interpretation of SUE is opened. SUE is the condition where, at macro level, the observed route flows do not change across days, while, at micro level, individuals change route. Only if random terms are unchanged no individual changes route. Transition flows at SUE are symmetric, i.e. the number of shifters from path i to path j equals the number of shifters from path j to path i. The insights are illustrated by numerical examples related to a two-link and a five-link network.

Sensitivity analysis for stochastic user equilibrium traffic assignment with constraints

A mathematical programming method for sensitivity analysis of the link-capacitated stochastic user equilibrium model is presented in this article. By the sensitivity matrices, the changes in the network flows can be easily obtained, while some links reach the capacity limits. By link-based stochastic user equilibrium model with link capacity constraints, it is possible to formulate an efficient algorithm for the sensitivity analysis. Two numerical examples are provided for demonstrating the correctness and implementability of the method finally. Since the link capacity is one of the constraints of the model, the method presented here can also be used for the basic model with other constraints.

Stochastic user equilibrium traffic assignment with equilibrated parking search routes

In this paper we define and formulate the concept of parking search routes (PSR) where a driver visits a sequence of parking locations until the first vacant parking spot is found and in doing so may account for (expected) parking probabilities. From there we define and formulate the stochastic user equilibrium (SUE) traffic assignment in which no driver, by unilaterally changing its PSR, can lower its perceived expected generalized costs. Recognizing the interdependency between PSR flows, travel times and parking probabilities, we propose a queuing model in order to compute endogenous parking probabilities accounting for these factors as well as maximum admissible search times. To solve the SUE assignment with equilibrated PSR we propose a solution algorithm, including a method for PSR choice set generation. The model is implemented and applied both to a number of experimental cases to verify its properties and to a real-life setting to illustrate its usefulness in parking-related studies.

Robust optimization of distance-based tolls in a network considering stochastic day to day dynamics

This paper investigates the nonlinear distance-based congestion pricing in a network considering stochastic day-to-day dynamics. After an implementation/adjustment of a congestion pricing scheme, the network flows in a certain period of days are not on an equilibrium state, thus it is problematic to take the equilibrium-based indexes as the pricing objective. Therefore, the concept of robust optimization is taken for the congestion toll determination problem, which takes into account the network performance of each day. First, a minimax model which minimizes the maximum regret on each day is proposed. Taking as a constraint of the minimax model, a path-based day to day dynamics model under stochastic user equilibrium (SUE) constraints is discussed in this paper. It is difficult to solve this minimax model by exact algorithms because of the implicity of the flow map function. Hence, a two-phase artificial bee colony algorithm is developed to solve the proposed minimax regret model, of which the first phase solves the minimal expected total travel cost for each day and the second phase handles the minimax robust optimization problem. Finally, a numerical example is conducted to validate the proposed models and methods.

Network capacity with probit-based stochastic user equilibrium problem.

Among different stochastic user equilibrium (SUE) traffic assignment models, the Logit-based stochastic user equilibrium (SUE) is extensively investigated by researchers. It is constantly formulated as the low-level problem to describe the drivers’ route choice behavior in bi-level problems such as network design, toll optimization et al. The Probit-based SUE model receives far less attention compared with Logit-based model albeit the assignment result is more consistent with drivers’ behavior. It is well-known that due to the identical and irrelevant alternative (IIA) assumption, the Logit-based SUE model is incapable to deal with route overlapping problem and cannot account for perception variance with respect to trips. This paper aims to explore the network capacity with Probit-based traffic assignment model and investigate the differences of it is with Logit-based SUE traffic assignment models. The network capacity is formulated as a bi-level programming where the up-level program is to maximize the network capacity through optimizing input parameters (O-D multiplies and signal splits) while the low-level program is the Logit-based or Probit-based SUE problem formulated to model the drivers’ route choice. A heuristic algorithm based on sensitivity analysis of SUE problem is detailed presented to solve the proposed bi-level program. Three numerical example networks are used to discuss the differences of network capacity between Logit-based SUE constraint and Probit-based SUE constraint. This study finds that while the network capacity show different results between Probit-based SUE and Logit-based SUE constraints, the variation pattern of network capacity with respect to increased level of travelers’ information for general network under the two type of SUE problems is the same, and with certain level of travelers’ information, both of them can achieve the same maximum network capacity.

Fixed-Point States of Day-to-Day Assignment Processes with State-Dependent Route Choice

Abstract Stochastic User Equilibrium (SUE) has been proposed to overcome the limitations of the perfect knowledge assumption of classical Wardrop’s user equilibrium. Imperfect knowledge of the network and the associated heterogeneity in the perception of route travel times, as it occurs in the absence of Advanced Traveller Information Systems (ATIS), justifies SUE providing strictly positive flows for all routes. SUE is at the same time the steady state of a day-to-day Markov assignment process under the assumption of state independent route choice, i.e. choice independent of the previously chosen route. Still in a context of absence of ATIS, state-dependent route choice represents a more realistic assumption: users exhibit inertia to change. Experimental evidence supports the assumption. The paper investigates the fixed-point states of Markov assignment processes with state-dependent route choice. These fixed points characterise a new concept of network equilibrium, referred to as State-Dependent Stochastic User Equilibrium (SDSUE), where if each user shifts from her current route to her newly chosen route the observed route flows do not change. The existence of SDSUE is guaranteed under usually satisfied conditions. A modified method of successive averages is proposed for computing SDSUE. An example related to the Braess network with logit route choice illustrates the theoretical insights. Numerical evidence is provided of a symmetry property of SDSUE transition flows whereby, at equilibrium, the number of users shifting from route i to route j equals the number of those shifting from route j to route i.