Weibull dependence stochastic traffic assignment model considering risk prone drivers
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摘要: 建立了考虑风险爱好驾驶人的相依Weibull随机交通分配(Weibull-DSA) 模型, 分析了感知等价路径负效用的Weibull边缘生存函数, 假设驾驶人总是选择感知等价路径负效用最小的路径到达目的地, 采用Copula方法构建了感知等价路径负效用的联合生存函数, 预测了路径选择概率; 设计了模型的迭代求解算法, 对模型进行了理论分析和数值验证; 研究了广州市交通调查获得的风险系数, 基于风险爱好和风险中立驾驶人, 比较了采用Weibull-DSA模型与经典的Logit-SUE和Weibit-SUE模型计算的路径选择概率、路段交通量、饱和度与系统总出行时间。计算结果表明: 随着风险系数的降低, 3种分配模型的交通系统总出行时间变大; 在风险中立情况下, 应用Weibull-DSA模型、Logit-SUE模型和Weibit-SUE模型计算得到每OD对的所有连接路径选择概率的最大差值, 分别为0.17、0.33、0.34, 在风险爱好情况下, 由3种模型得到的最大差值分别为0.20、0.36、0.41, 因此, 采用Weibull-DSA模型计算得到的不同路径选择概率的最大差值明显小于经典模型计算得到的最大差值; 相对于风险中立情况, 风险系数使得每OD对的所有连接路径选择概率的最大差值变大; 无论是风险爱好还是风险中立驾驶人, 采用Logit-SUE和Weibit-SUE模型计算得到的路段饱和度均小于0.9, 采用Weibull-DSA模型计算得到路段饱和度大于0.9;与经典模型计算结果不同, 采用Weibull-DSA模型得到的不同路径选择概率的最大差值相差较小, 一些路径获得更多交通量, 使得路径中通行能力最小的路段的饱和度大于0.9, 这一特征给出了城市路网中部分瓶颈路段拥堵现象一个新的解释。Abstract: A Weibull dependence stochastic traffic assignment (Weibull-DSA) model considering risk prone drivers was established. The Weibull marginal survival function of perceived equivalent route disutility was analyzed.It was assumed that travelers always chose the routes with the minimized expected perceived equivalent route disutility to reach their destinations.Thejoint survival function of perceived equivalent route disutility was constructed by using Copula method, and the route choice probability was predicted.An iterative solution algorithm was designed for the model, and the theoretical analysis and numerical verification of the model were carried out.Risk coefficient obtained from the traffic survey in Guangzhou was analyzed.The route choice probabilities, road section traffic volumes, saturation degrees and total travel times were calculated by using Weibull-DSA model, classic Logit stochastic user equilibrium (LogitSUE) model and Weibit stochastic user equilibrium (Weibit-SUE) model under the assumption of risk prone and risk neutral drivers, respectively.Calculation result shows that as the risk coefficient becomes smaller, the total travel times of traffic system for all three kinds of traffic assignment models become larger.Under the risk neutral condition, the maximum differences among the route choice probabilities of all alternative routes connecting each OD pair are 0.17, 0.33, 0.34, respectively calculated by using Weibull-DSA model, Logit-SUE and Weibit-SUE model.Similarly, under the risk prone condition, the maximum differences calculated by the three models are 0.20, 0.36 and 0.41, respectively.Therefore, the maximum difference of different route choice probabilities calculated by Weibull-DSA model is obviously less than the maximum differences obtained by two classical models.Compared with the risk neutral situation, the risk coefficient increases the maximum difference of the route choice probabilities of all alternative routes connecting each OD pair.For either risk prone or risk neutral drivers, the saturation degrees of each road section calculated by Logit-SUE and Weibit-SUE models are all less than 0.9, whereas the saturation degrees obtained by Weibull-DSA model are more than 0.9.Unlike the calculation results of classical models, the maximum difference of route choice probabilities obtained by Weibull-DSA model is smaller, some routes get more traffic volumes, which makes the saturation degrees of road sections with the smallest capacity in the route are greater than 0.9.This feature gives a new explanation for the congestion phenomenon at some bottlenecks in the urban road network.
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道路交叉口的通行能力影响着整个道路网的通行能力, 由于车辆的转向而引起车流之间的冲突、交汇、分流等车流运行行为, 使交叉口的交通特性较为复杂, 观测数据较为困难, 因此, 研究无控平面交叉口的通行能力成为道路通行能力研究的难点。许多学者通过改进主路交通流车头时距的分布函数, 用能够反映实际主路车头时距的分布函数代替负指数分布函数, 进而得到了许多不同的理论模型[2~7]。李文权、王炜[8]拓广了支路车型为多种混合车型的通行能力模型(以下简称为M-混合通行能力模型), 但仍然是假设主路车流车头时距服从M3分布。随后, 常玉林、王炜[9]得到了主路车流车头时距服从二阶Erlang分布下的支路通行能力模型见式(1), 但支路车流仅局限为单一的理想车型
对于车头时距服从Erlang分布的支路多车型混合车流情况, 由于其模型公式的推导比较复杂, 以前大都采取数值模拟的方法, 理论上比较少见。据此, 本文通过将单一车型交通流的假设拓广为由r种车型构成的混合车流, 推导主路车流车头时距服从二阶Erlang分布下支路混合车流的通行能力模型, 使式(1)变成无控交叉口混合车流通行能力模型的推论, 并结合实际例子进行了对比分析。
1. 基本假设
考虑如下一种最简单的情形: 在无控交叉口的冲突车流为两支直行车流, 其中主路车流车头时距服从二阶Erlang分布, 支路车流由1型车辆、2型车辆、…、r型车辆的r种车型构成, r种车型的构成比例为q1: q2: …: qr, 并且q1+q2+…qr=1。驾驶同种类型车辆的驾驶员假设行为一致, 其通过无控交叉口冲突区时遵循可接受间歇理论。不同车型的临界间隙和随车时距不同, 假设k型车辆的临界间隙和随车时距分别为tck和tfk, k=1, 2, …, r, 不同类型车辆到达交叉口的随机事件相互独立, 支路有充分多的车辆等待通过交叉口, 可以容纳无限多的车辆排队。假设无控交叉口主路车流车辆间的车头时距为h, 根据可接受间歇理论, 当
时, 允许支路一辆k型车通过交叉口冲突区; 当
时, 允许一辆k型车排头, 后面紧跟另一辆j型车通过交叉口冲突区; 当
时, 允许支路一辆k型车排头, 后面紧跟n1辆1型车辆、n2辆2型车辆、…、nr辆r型车辆通过交叉口冲突区。
2. 无控交叉口混合车流的通行能力
由于支路车流由r种车型混合而成, 每一种车型到达交叉口的随机事件相互独立, 因此在分析支路车流通行能力之前, 首先要对支路车辆可能出现的排队构形及其概率进行分析。
2.1 支路混合车流的排队构形及其概率
由于考虑支路车流中1型车辆、2型车辆、…、r型车辆r种车型构成的比例为q1: q2: …: qr, 并且q1+q2+…+qr=1, 因此对于每一时刻第k型车处于排队队首的概率为qk, 其中k=1, 2, …r, 也就是队长为1的所有排队构形为1型车辆、2型车辆、…、r型车辆; 队长为2的所有排队构形为: 1-1、1-2、…、1-r; 2-1、2-2、…、2-r; …; r-1、r-2、…、r-r, 其中1-r表示1型车辆后面紧跟着r型车辆, 其他类似。由不同类型车辆到达交叉口的随机事件相互独立而知, 队长为2以上各排队构形的相应概率为: q12、q1q2、…、q1qr; q2q1、q22、…、q2qr; …; qrq1、qrq2、…、qr2, 不难验证以上各项概率值的和为
一般地, 当队长为n时的排队构形为: k型车辆~(队长为n-1的排队构形), 其中k=1, 2, …, r。队长为n-1的排队构形中有n1辆1型车辆、n2辆2型车辆、…、nr辆r型车辆的不同排队构形是一个推广的贝努利试验, 因而服从多项分布, 其概率为
(n−1)!n1!n2!⋯nr!qn11qn22⋯qnrr , 其中n1+n2+…+nr=n-1。于是k型车辆位于排队队首, 其后面队长为n-1的排队构形中有n1辆1型车辆、n2辆2型车辆、…、nr辆r型车辆的概率为
不难验证
2.2 支路车辆通过特定n辆车的概率
设主路车流车头时距服从参数为λ的二阶Erlang分布, 则主路车流的交通流率为λ/2(veh/s)、参数为λ的二阶Erlang分布, 其概率分布函数为
于是, 主路车头时距能够通过支路k型车辆位于排队队首, 其后面队长为n-1的排队构形中有n1辆1型车辆、n2辆2型车辆、…、nr辆r型车辆的概率为(其中n1+n2+…+nr=n—1;k=1, 2, …, r)
2.3 支路车辆通过任意n辆车的概率
根据式(5)与式(7)可以知道, 无控交叉口主路车头时距能够保证一次通过支路k型车辆位于排队队首, 其后面队长为n-1的排队构形中有n1辆1型车辆、n2辆2型车辆、…、nr辆r型车辆的概率为
于是, 无控交叉口主路车流车头时距能够保证一次通过支路车辆n辆车的概率为
2.4 支路混合车流的通行能力
通过分析, 一个主路间隙内支路混合交通流通过无控交叉口车辆数的平均值(数学期望值)为
由于主路车流的交通流率为λ/2(veh/s), 于是1 s内主路车流能够提供λ/2个间隙, 从而支路混合车流通过无控交叉口的通行能力(veh/s)为
3. 实例分析
3.1 交叉口的观测数据
所观测具体交叉口为一个2×2路十字无控交叉口[8], 交通流的构成按大、中、小3种车辆划分, 交通流基本上没有横向干扰。交叉口各进口引道上的交通组成比例为大型车: 中型车: 小型车=22:32:46;转向比例为左转车辆: 右转车辆: 直行车辆=16:18:66;二阶Erlang分布的参数λ由主路流率观测数据的均值m和方差s2计算, λ=m/s2。交叉口不同类型车辆的临界间歇、随车时距、当量车换算系数见表 1。
表 1 不同车型的临界间歇和随车时距Table 1. Critical intermittences and follow time-gaps of different vehicle types
研究交叉口8个15 min观测时间段的观测交通量及其平均延误和实际通行能力见表 2。
表 2 八个15 min观测时间段的观测结果Table 2. Observation results in eight observation continuing 15 min
3.2 理论模型的对比分析
下面分别以式(1)模型、M-混合通行能力模型[8]、美国1994年通行能力手册推荐的Siegloch模型[2]和本文推导的混合通行能力模型式(11)按照相同的折减方法[1, 2]计算实例交叉口的理论模型通行能力, 然后结合表 2的实际观测通行能力计算绝对误差、相对误差, 结果见表 3。
表 3 八个15 min观测时间段的理论计算结果及误差Table 3. Theory results and errors in eight observation continuing 15 min
用Siegloch模型、式(1)模型、M-混合通行能力模型和本文推导的混合通行能力模型所计算的通行能力的最大绝对误差分别为: 1 064、1 059、939、818veh/h; 平均相对百分误差分别为26.271 8%、25.933 8%、17.361 2%、16.603 8%。由此可见本文推导的理论模型更切合实际情况。
4. 结语
本文对无控交叉口主路车流车头时距服从二阶Erlang分布, 支路有多种车型构成的混合交通流的通行能力模型进行了理论推导, 得到了式(11)的理论模型。当交叉口的交通流为单一车流, 而主路车流车头时距服从二阶Erlang分布时, 也就是tc1=tc2=…=tcr=tc, tf1=tf2=…=tfr=tf, r=1, 式(11)便退化为传统的单一车型交通流的通行能力理论模型式(1), 因此理论模型式(11)发展了无控交叉口的通行能力理论; 并且通过与已有模型与实际观测通行能力的对比分析可以说明, 通过改进主路车流车头时距以及把支路上的车型从单一车型拓广为多种车型能使理论模型通行能力与实际观测通行能力的接近程度更好。
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图 2 连接OD对1-5的6条路径的路径选择概率
Figure 2. Route choice probabilities of six alternative routes connecting OD pair 1-5
图 3 连接OD对5-1的6条路径的路径选择概率
Figure 3. Route choice probabilities of six alternative routes connecting OD pair 5-1
图 4 连接OD对3-7的6条路径的路径选择概率
Figure 4. Route choice probabilities of six alternative routes connecting OD pair 3-7
图 5 连接OD对7-3的6条路径的路径选择概率
Figure 5. Route choice probabilities of six alternative routes connecting OD pair 7-3
图 6 风险中立情况下Logit-SUE模型得到的路段饱和度
Figure 6. Saturation degrees of road sections resulting from Logit-SUE model under risk neutral condition
图 7 风险中立情况下Weibit-SUE模型得到的路段饱和度
Figure 7. Saturation degrees of road sections resulting from Weibit-SUE model under risk neutral condition
图 8 风险中立情况下Weibull-SUE模型得到的路段饱和度
Figure 8. Saturation degrees of road sections resulting from Weibull-SUE model under risk neutral condition
图 9 风险爱好情况下Logit-SUE模型得到的路段饱和度
Figure 9. Saturation degrees of road sections resulting from Logit-SUE model under risk prone condition
图 10 风险爱好情况下Weibit-SUE模型得到的路段饱和度
Figure 10. Saturation degrees of road sections resulting from Weibit-SUE model under risk prone condition
图 11 风险爱好情况下Weibull-SUE模型得到的路段饱和度
Figure 11. Saturation degrees of road sections resulting from Weibull-SUE model under risk prone condition
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