An Enhanced Dynamic User Optimal Passenger Flow Assignment Model for Metro Networks

By considering the difference between a car driver’s route choice behavior on the road and a passenger’s route choice behavior in urban rail transit (URT), this paper proposes an enhanced Dynamic User Optimal (DUO) passenger flow assignment model for metro networks. To capture realistic URT phenomena, the model has integrated the train operation disturbance constraint. Real passenger and train data are used to verify the proposed model and algorithm. The results indicate that the DUO-based model is more suitable for describing passenger route choice behavior under uncertain conditions compared to a static model. Moreover, this paper found that passengers under oversaturated conditions are more sensitive to train operation disturbances compared to undersaturated passengers.


Introduction
Although the car driver's dynamic route choice behavior has been studied for several decades, little attention has been paid to passenger's route choice behavior in urban rail transit [1].At present, most previous studies do not consider dynamic features when passengers make decisions in urban rail transit (URT).The limitations of these models include the inability of modeling passengers' departure times and realtime congestion.
The earliest known work related to dynamic features is the bottleneck model proposed by Vickrey [2].After several decades of developments, many models have been proposed.For example, Yang and Jiang [3] proposed an enhanced route choice model based on cumulative prospect theory to describe dynamic passenger choice behavior.By considering queue spillback, Zhou et al. [4] developed a model to describe dynamic passenger assignment behavior based on AFC Data.However, these models are suitable only for simple networks.Therefore, researchers began to search for new theories to describe passenger route choice behavior.Among them, the optimal dynamic models received considerable attention.Optimal dynamic route choice principles can be classified into two categories: the Dynamic User Optimal (DUO) principle [5][6][7] and the Dynamic System Optimal (DSO) principle [8][9][10].Obviously, managers prefer traffic assignment, which is consistent with the DSO principle.However, DUO can describe the actual passenger distribution in URT more accurately.
DUO is used to represent equilibrium state in a timevarying URT network.This equilibrium can be described as follows: for each instant, travel costs of all used routes between each origin-destination (OD) pair are equal and are less than the unused routes.Dafermos and Sparrow [11] found that such equilibrium ensures that "for each instant, none of the travelers could decrease his/her travel cost by unilaterally changing his/her route." To develop mathematical models that describe DUO principle, several previous studies had been conducted by researchers.Since the 1990s, considerable attention has been paid to variational inequality (VI) for analyzing DTA problems based on DUO.Friesz et al. [12] first proposed variational inequality to describe DUO conditions.However, this model is very difficult to solve.After several years of research and based on monotonic path cost function, Friesz and Mookherjee [13] proposed a projection algorithm that converges to a DUO solution.
In the aforementioned studies, it is obvious to note that DUO principle is widely used to develop models of road traffic assignment.However, these models cannot be used for describing passenger route choice behavior directly because there are some important differences between them.
(1) Difference of Research Object.In the field of road traffic assignment, we pay attention to a driver's route choice behavior, while, in rail transit, we focus on the choice behavior of passengers.Drivers change their route in an intersection, while passengers change their route in a transfer station.
(2) Difference between Traffic Flow and Passenger Flow.For road traffic, vehicle speed is closely related to the density of traffic flow, while train operation is predetermined by the timetable and experiences little impact from passenger density.
(3) Difference of Path Impedance.In the field of road traffic, Path impedance usually consists of travel time, road condition, and mileage.Meanwhile, for URT, path impedance mainly consists of travel time, comfort, transfer time, train headway, and so forth.
In fact, with the continuous increase in passenger volume in URT, the frequency of train operation disturbance also increases [14][15][16].To retain the advantage of DUO models, passengers' reaction to the disturbance should be considered.In this paper, the DUO problem is formulated with a disturbance constraint to capture the realistic URT phenomena.Furthermore, we proposed a feasible algorithm based on Frank-Wolf 's one to solve this model.
The remainder of this paper is organized as follows: the formulation of network and impedance function is presented in Section 2. Section 3 proposed a passenger flow assignment model and algorithm.A numerical experiment is given in Section 4. Finally, conclusions are provided in Section 5.

Model Formulation
2.1.Network Representation.This study considers an urban metro network that contains several stations.Nodes (set N) and links (set L) are extracted from the practical metro network.The vector (, ) represents the directed links from station  to station .Three different lines cross with each other to illustrate a simple network, as shown in Figure 1.Each transfer station consists of several ordinary stations depending on the number of linked lines.For example, transfer station A is composed of node 302 (located on line 3) and node 101 (located on line 1).Then, direct links (302, 101) and (101, 302) represent transfer activity between these two nodes.

Assumptions.
To better model dynamic passenger route choice behavior in URT, three basic assumptions are presented as follows: (1) Passengers have perfect knowledge of the travel conditions.That is, passengers can change their original path using real-time information about the travel scenario.(2) Passengers will not cancel their trip due to the impact of train operation disturbance.That is, no passengers will leave the subway system until they arrive at the destination station.
(3) Transit lines do not have capacity limits; that is, all passengers who are waiting on the platform can board the train.

Impedance Function.
Traditional static passenger assignment models (either deterministic or stochastic) assume that the assigned passenger flow exists in every link of the path between the OD pair.The impedance of the path is generally constant and will not change with passenger flow distribution.Obviously, this assumption is not accurate enough.Therefore, in this paper, we focus on the impedance of links instead of the whole path.Moreover, the impedance is changed with the number of passengers who are traveling through the corresponding link.
The main component of path impedance is travel time.Therefore, in this paper, the impedance of each path contains four parts: (1) travel time on the train; (2) transfer time; (3) waiting time; and (4) overload delays (1) Travel Time on the Train.Travel time on the train contains train section running time and dwell time, which are predetermined by the timetable.Obviously, the timetable is not affected by the volume of passenger flows.Therefore, travel time on the train can be calculated as follows: where   represents train running time from node  to node  and   represents train dwell time.  represents a path , which contains a number of nodes.
(2) Transfer Time.Transfer activity occurs when passengers need to change from one line to another.With the development of URT in China, more than fifty percent of passengers need to transfer during their trips.Thus, transfer activity in path finding turns out to be an important component of passenger travel.
In general, transfer time is usually associated with the passenger walking speed and the length of the transfer channel.Therefore, transfer time can be calculated as follows: where    represents the transfer length from node  to node  and V  denotes the average walking speed.A penalty needs to be considered for transfer activity to indicate extra transfer cost.Therefore, transfer time can be denoted as follows:  ⋅  trans , where  represents the penalty coefficient.
(3) Waiting Time.Passenger waiting time is related to the train headway and the time that passengers arrive at the platform.According to a previous study [17], the randomness of a passenger's departure time is greatly reduced when the train headway is greater than 12 minutes.They tend to arrange their travel plans according to the train departure time.In contrast, the travel is often random when the interval of the train is less than 12 minutes.That is, as shown in Figure 2, passenger waiting time tends to be a constant when the train interval is long (more than 12 minutes), and passenger waiting time tends to be half of the train headway when the train interval is short (less than 12 minutes).Therefore, the waiting time can be calculated as follows: where   represents train headway of line .
(4) Overload Delays.As mentioned above, the average passenger waiting time can be calculated as half of train headway.However, under oversaturated conditions, not all passengers waiting on the platform are able to board the train, and some passengers are left behind.For example, passengers crowded on the platform may cause the platform screen door not to close.Therefore, overload delays are complicated and difficult to calculate.BPR [18] formulation is applied in this study to estimate overload delays as follows: where    () is the overload delays for link  at time  and  represents train capacity. and  are parameters, which can be obtained by actual survey.

Model of Passenger Flow Assignment Based on DUO.
The DUO assignment model is well known as a general model that consistently unifies dynamic features and a Wardrop equilibrium.According to the definition of DUO, for each node and instant, impedances of all used paths between each OD pair are equal, which equal the minimum instantaneous impedance.Meanwhile, the instantaneous impedances of all unused paths are more than the used ones.
To capture the disturbance of train operation in an actual network, we developed a DUO-based model.In this paper, we divide the study period into several equal time intervals denoted by .According to assumption (2), passengers will not cancel the trip when they enter the subway system.Therefore, all inflows into any node must be equal to all outflows from this node, and the flow conservation can be formulated as follows: where   represents the set of inflows into node  and   represents the set of outflows from node .If the node is destination or origin stations, the above equation should be extended as below: where   and   are binary parameters.If node  is an origin station,   = 1; otherwise   = 0. Similarly, if node  is a destination station,   = 1; otherwise,   = 0.The next task is to derive a mathematically tractable equation for temporarily calculating the passengers traveling through each directed link at any time.We can obtain the equation by flow conservation, given as follows: where    () denotes the number of passengers who are traveling through direct link  to destination .  () denotes the cumulative number of passengers who are traveling through direct link .
Inflow into link  during interval  can be calculated as Similarly, outflow from link  during interval  can be obtained by Moreover, nonnegative constraints must be satisfied: In a URT practical network, passengers can board or alight from the train only at the stations (nodes).That is, no passengers could leave the system in direct links.When time is discretized,    can be calculated as follows: In this paper, passengers traveling through the direct links obey the First-In-First-Out (FIFO) principle [19].In other words, passengers who enter the direct link cannot overtake each other.We denote the impedance of direct link  at time  as   ().If passengers enter the direct link at time , then they will leave the link at time  +   ().Similarly, if passengers enter the direct link at time  + Δ, then they will leave the link at time  + Δ +   ( + Δ).Therefore, FIFO principle can be represented as follows: Disturbance of the train operation can affect passenger's predetermined travel plan.For example, it is possible that passenger waiting time is greater than they estimated before departure due to a disturbance.In our model, we add the constraint that limits passengers entering the direct links when they are affected by train disturbance: where  represents the set of stations influenced by disturbance propagation.
According to the DUO definition by a previous study [20], passengers dynamically change their route in each travel node.It is obvious that a passenger's decision making is dependent on the instantaneous impedance in the network.As we know, the impedance of the direct link is related to the number of passengers who are traveling through the link.Therefore, the objective function of the proposed model is given by 3.2.Solution.Overall, due to the complexity of the URT network, the proposed passenger flow assignment model based on DUO is difficult to solve by traditional commercial solvers.The Frank-Wolf algorithm, which was first proposed by Frank and Wolfe, is worth considering [21].This algorithm has been used to address traffic problems since the 1970s, and it is still considered an effective way to solve DUO problems today.The proposed algorithm in this paper is similar to previous ones, but the major differences are their definition of impedance and disturbance of train operation.The Frank-Wolf based algorithm for the DUO problem in URT is outlined as follows.
Step 2. Determine the impedance of each of the links based on the passengers who are traveling through the corresponding link; if there is disturbance of train operation, then modify the corresponding   ().
Step 3. Use an all-or-nothing algorithm to calculate inflow rates for each link in the network, which is a feasible solution.
Step 4. Use (11) to calculate the number of passengers who are traveling through directed link at time , and use (4) to calculate the impedance of each link.Then, use (5) to calculate outflow of each link at time .According to the conservation principle, passenger flows are assigned in the network based on an all-or-nothing algorithm.
Step 5. Use equation    () = 0 to modify the influence of train disturbance.Then, calculate the instantaneous impedance for each link as follows: where    () and    () denote the minimum instantaneous impedance from nodes  to  and nodes  to , respectively.
Step 6.According to the conservation principle, the inflow to be assigned at each node is ∫ Use an allor-nothing algorithm to calculate the initial feasible assigned inflow (denoted as     ) for each link in the network.Then, calculate the optimal convergent step length as follows: Step 7. Update the assigned inflow for each link as follows: Step 2 / ∑     () ≤ , then stop the algorithm; otherwise, set  =  + 1, and go to Step 5.
The above Frank-Wolfe solution can be directly used to solve the proposed passenger flow assignment model.Note that  is a predetermined precision parameter.

Case Study
4.1.Experiments in a Practical Network.In this section, a numerical example is presented to illustrate the performance of the proposed model and algorithm in a tested network as shown in Figure 3.
According to Section 2, we divided each transfer station into two individual platform nodes locating on different lines.Then, the grid network is extracted from the practical tested rail network as shown in Figure 4.This network consists of 24 nodes and 44 direct links.Note that direct links between these two nodes represent transfer activity.According to the above section, we consider a penalty for each transfer activity to indicate extra cost.The travel times of each link are randomly generated as shown in Table 1.The value of parameters , , , and c are set to 2, 1.5, 1.5, and 200, respectively.Assume that train disturbance occurs at direct link (12,14) at time interval 7, which causes a 15-minute train delay.To obtain the solution for the given case, the proposed model and Frank-Wolfe algorithm are implemented in a personal computer with 4.00 GB Memory and an Intel Core i5 processor.
All the networks were empty initially.For descriptive convenience, nodes 11 and 31 are selected as the origin and destination station, respectively.The study period is set to 50 minutes, which is from 7:00 to 7:50.The calculation results are shown in Table 2.
For intervals 1-3, all passengers select path 2. This indicates that no congestion and train operation disturbance occurred during the beginning intervals.When congestion exists in path 1, it is very easy to see that part of the passengers select path 2.Moreover, in DUO, passengers make decisions at each reached node dynamically.This implies that passengers may change their original path using real-time information at any nodes.
According to the assumption mentioned above, train disturbance occurs at direct link (12,14) at the beginning of interval 7.That is, the impedance of path 2 is suddenly increased.The selected result above shows that 110 passengers are traveling through directed link (13,12) at time 7.This indicates that some of the passengers who initially choose path 1, which is influenced by train operation disturbance, suddenly choose path 2 instead of path 1.     [22] is referred to as M2.The test network is shown in Figure 6.Nodes 1 and 8 are selected as the origin and destination station, respectively.The travel time of each link without overload delays is set to 5 minutes.In the following, M1 and M2 will be tested in four types of practical scenarios: (1) Oversaturated and disturbance of train operation conditions occurring at direct link (3,4).
Two valid paths are obtained by the Dial algorithm, as shown in Table 3.The calculated results of M1 and M2 under different scenarios are shown in Table 4.The selection results of these two models are consistent in scenarios 3 and 4. That is, M1 is similar to M2 when the trains are operating strictly according to the predetermined timetable.
Varying degrees of separation exist in scenarios 1 and 2. According to the assumption, train operation disturbance occurs at direct link (3,4).The impedance of path 2 is suddenly increased.Passengers who originally select path 1 are confronted with the disturbance of train delay.Therefore, a certain part of the passengers will change their original path.The numerical result shows that M2 does not consider the influence of train operation disturbance when passengers make decisions.M1 provides the dynamically calculated process, which can overcome the shortcomings of M2 to some extent.Therefore, M1 is better than M2 for describing passenger route choice behavior when train operation disturbance occurs.

Conclusions
This paper proposed an enhanced passenger flow assignment model for metro networks.Compared with the traditional static model, the DUO-based model is more suitable for describing dynamic passenger distribution during rush hour.In addition, passenger route choice behavior in URT is different from car driver's route choice behavior on the road.Therefore, the method of determining the impedance in the metro network was improved in this study.
A more favorable characteristic for the DUO-based model was found: train operation disturbance.Indeed, this paper found that passengers experiencing oversaturated condition are more sensitive to disturbances than those experiencing undersaturated conditions.Furthermore, the DUO-based model proposed in this paper is robust and may be applied to oversaturated conditions for URT networks.
In terms of future research, it is necessary to calibrate relevant parameters under different travel purposes and mentalities.Moreover, randomness of passenger travel time should also be considered.

Figure 1 :
Figure 1: The representation of a simple network.

Figure 2 :
Figure 2: Relationships between the average waiting time and the headway.

Figure 4 :
Figure 4: A grid network for illustrating solution process.

Figure 6 :
Figure 6: Tested network for comparative analysis of traditional and DUO approaches.

Table 1 :
The travel time of each link without overload delays (min).

Table 2 :
The assigned passengers for directed links during each interval.