收费站通道开放数量优化的动态瓶颈模型分析

Dynamic model on optimal open toll gates number of the highway

本文建立数学模型,在节约高速公路收费站的运营成本和减少出行者排队延误之间做出平衡,根据变化的交通量动态调节收费通道开放的数量。基于动态瓶颈模型,建立了考虑用户均衡原则的高速公路开放收费通道数量优化模型。研究分析了收费口开放数量与系统费用、出行者费用(延误费用)以及运营费用之间的关系,以及在系统最优时,出行者费用与收费站运营费用之间的比例关系。研究得出当出行需求变化时,系统最优收费通道数量不同。因此根据交通需求的变化,为了达到系统费用最低,应开放与之相适应的收费通道数量。其次,本文分析了建设收费口数量与总费用之间的关系,发现建设过多的收费口数量不一定能降低系统的总费用,有时会造成系统总费用的上升。最后,研究分析了模型中的超负荷运营费用系数和夜间运营费用系数对模型具有较好的稳定性。

Less attention has been received for highway congestion because of the limited toll station capacity on holidays or large-scale activities, such as feast days, conferences or exhibitions. Most of travelers choose the similar departure time on the same highway and peak-load demand lead to queues at toll station areas. The capacity of the toll station will be increased if more toll gates are opened. Accordingly, the travel time will be decreased and operational cost of the toll station will be raised. On the contrary, the travel time and the queue will be amplified without opening enough toll gates. In this paper, the optimal number of open toll gates in highway is analyzed based on the dynamic bottleneck model with time-varying capacities under user equilibrium. The research of bottleneck congestion and departure time choice can be traced to Vickrey’s (1969) study that considered a bottleneck road connecting two ends H (home) and W (work place) with a fixed capacity. If the arrival rate at the bottleneck exceeds capacity, a queue develops. All drivers wish to arrive at work at the same time. Departure pattern based on an optimization rationale for individual commuters can be derived. Travelers are assumed to balance the trade-off between schedule delay and travel time in a deterministic way. That is to say, if she leaves early, she faces no queue but arrives at work inconveniently early;if she leaves so as to arrive on time, she faces a long queue;and if she leaves late, she faces no queue but arrives inconveniently late. An equilibrium is obtained when the queue length over time is such that no driver can reduce his trip price by changing departure time. We assume that N commuters live in H at night and take part in conferences or exhibitions in W in the day time. In the morning, individuals have a common preferred arrivial time t* at conferences or exhibitions at W and early or late arrival will be penalized. The capacity of the bottleneck depends on how many toll gates are open. The first individual choose departure time at ta, and the last individual departure at tb. There are z+2 intervals and the capacity for each time interval is constant. For example, for interval ta to t0 the capacity is s0, s0=j×s , j is number of open toll gates for time ta to t0, s is capactiy for each open gate. For interval tz to tb the capacity is se, es=n×s , n is number of open toll gates for timetz to tb. For other intervals (from ti-1 to ti), toll gates capacity can be described as si=θi-1×s,(i=1,……z),θi-1 is number of open toll gates for time ti-1 to ti.analytic solutions for the Schedual Delay Cost (SDC) can be gained when the capacity of the toll station is time-varying which depends on how many toll gates are open. The Operation Cost (OC) equals to operation cost for each gate multiply by the number of the open gates. Operation Cost (OC) has two impact factors.(1) The individuals who depart at the beginning and end of the rush hour incure only schedual delay cost without queue at toll station, which must be equal in user equilibrium. Let the start time of queue be ta and end time be tb, if the start time ta is too early, the more toll collectors work earlier cause more operation cost compared to daytime, that is because the toll collectors working in night or before dawn get more allowance.(2) In the rush hour, the travel cost will be decreased by opening more toll gates under equilibrium toll principle. If the open toll gates at peak period are much larger than average open toll gates, emergent operation scheme should be carried out which also result in additional cost. The System Cost (SC) is the addition of SDC and OC. How to get the least System Cost (SC) by designing to open more toll gates at peak period and fewer gates at off peak period is discussed in this paper. Several conclusions are drawn as follows.Firstly, based on the bottleneck model with time-varing capacities, the user equilibrium analytic solutions for the Schedual Delay Cost, Operation Cost and the System Cost are given. The analytic solution for Operation Cost is proved to be a constant when the toll gate operation cost is uniform for all time intervals from ta to tb, which is direct proportional to the total travel demand and inversely proportional to the each toll gate capacity, however, is independent of ta to tb. In fact, the toll station operation cost is not uniform, therefore the overload operating parameter τ and nighttime operatin parameter μ are introduced in the Operation Cost model. Secondly, the relationship between open gate number and SC, SDC and OC are investigated in this paper. The optimal open toll gates number are caculated by minmizing the System Cost. Assumed that the open number of the toll gates is uniform for all time intervals, the open toll gates number differs with the travel demand N and the optimal open gates increase with more travel demand by minmizing the total System Cost. The ratios between Schuedual Delay Cost and Operation Cost under different traffic demand are compared: There is an increasing trend for the ratio of the Operation Cost while there is a decreasing trend for the ratio of the Schedual Delay Cost with the increase of the travel demand. Thirdly, the optimal open gates number solution which includes open gates number and division of time intervals is varied with traffic demand. Thus, in order to minimize System Cost, open toll gates number within time intervals should be carried out corresponding to different scale activities. Fourthly, the results show that constructing more toll gates facilities can not always decrease System Cost. On the contrary, System Cost increases because of quicker improving on Operation Cost compared to Schedule Delay Cost saving. The decision on toll gates construction number should depend on highway design capacity. Finally, parameter robustness analysis for the overload operating parameter τ and nighttime operatin parameter μ are discussed. Value changes for these two parameters have no impact on the optimal solution. Therefore, the parameters τ and μ have better stability. The model can also be used to decide the optimal ratio between Electronic Toll Collection (ETC) and Manual Toll Collection (MTC), and the optimal toll gates construction number.