王昊1,2,3,秦严严1,2,3
(1.城市智能交通江苏省重点实验室(东南大学), 南京 210096; 2.现代城市交通技术江苏高校协同创新中心 (东南大学), 南京 210096; 3. 东南大学 交通学院, 南京 210096)
摘要:
针对网联车与普通车构成的混合交通流不稳定性问题,提出一种网联车混合交通流渐进稳定性解析方法. 基于传递函数理论,应用跟驰模型推导扰动在交通流中传播时的传递函数,并建立不同网联车比例下的混合交通流渐进稳定性解析框架. 选取智能驾驶模型(intelligent driver model, IDM)与优化速度模型(optimal velocity model,OVM)分别作为网联车与普通车的跟驰模型,进行混合交通流渐进稳定性案例分析,并进行小扰动下的数值仿真. 研究结果表明:所建立的混合交通流渐进稳定性解析框架可计算得到关于网联车比例与平衡态速度的混合交通流稳定域;当平衡态速度大于21.5 m/s时,混合交通流可在任意网联车比例下稳定,当网联车比例大于0.63时,混合交通流可在任意平衡态速度下稳定;混合交通流稳定性数值仿真实验验证了理论解析的正确性. 所建立的网联车混合交通流渐进稳定性解析框架适用于不同跟驰模型的选取,能够用于分析真车实验条件下网联车对交通流稳定性的影响.
关键词: 交通流 渐进稳定性 解析研究 网联车辆 传递函数
DOI:10.11918/j.issn.0367-6234.201711066
分类号:U491.112
文献标识码:A
基金项目:国家自然科学基金(3,1); 江苏省研究生科研与实践创新计划(KYCX17_0146); 东南大学优秀博士学位论文培育基金(YBJJ1792)
Asymptotic stability analysis of traffic flow mixed with connected vehicles
WANG Hao1,2,3,QIN Yanyan1,2,3
(1. Jiangsu Key Laboratory of Urban ITS (Southeast University), Nanjing 210096, China; 2. Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies (Southeast University), Nanjing 210096, China; 3. School of Transportation, Southeast University, Nanjing 210096, China)
Abstract:
Focusing on the instability of mixed traffic flow including connected vehicles and traditional vehicles, this paper proposes an analytical method for the asymptotic stability of the mixed traffic flow with connected vehicles. Based on the transfer function theory, the car-following models were used to derive the transfer function of disturbances spreading in traffic flow. Then the analytical framework of asymptotic stability of mixed traffic flow was built under different proportions of connected vehicles. The intelligent driver model (IDM) and optimal velocity model (OVM) were selected as car-following models for connected vehicles and traditional vehicles, respectively. Subsequently, case analysis of asymptotic stability of mixed traffic flow was conducted and numerical simulations were performed under small disturbances. Results showed that the proposed analytical framework of asymptotic stability of the mixed traffic flow can be used to calculate the stability region of the mixed traffic flow related to connected vehicle proportion and equilibrium velocity. When the equilibrium velocity was larger than 21.5 m·s-1, the mixed traffic flow was stable under any connected vehicle proportion. When the connected vehicle proportion was larger than 0.63, the mixed traffic flow was stable under any equilibrium velocity. Moreover, the numerical simulation experiments of mixed traffic flow validated the soundness of the theoretical analysis. The analytical framework of asymptotic stability of mixed traffic flow can be applied to different car-following models. It can also be used to analyze impacts of connected vehicles on traffic flow stability under the condition of real experimental tests.
Key words: traffic flow asymptotic stability analytical study connected vehicles transfer function
王昊, 秦严严. 网联车混合交通流渐进稳定性解析方法[J]. 哈尔滨工业大学学报, 2019, 51(3): 88-91. DOI: 10.11918/j.issn.0367-6234.201711066.
![复制到剪切板](http://hit.alljournals.cn/html/hitxb_cn/2019/3/images/copy.gif)
WANG Hao, QIN Yanyan. Asymptotic stability analysis of traffic flow mixed with connected vehicles[J]. Journal of Harbin Institute of Technology, 2019, 51(3): 88-91. DOI: 10.11918/j.issn.0367-6234.201711066.
![复制到剪切板](http://hit.alljournals.cn/html/hitxb_cn/2019/3/images/copy.gif)
基金项目 国家自然科学基金(51478113,51878161);江苏省研究生科研与实践创新计划(KYCX17_0146);东南大学优秀博士学位论文培育基金(YBJJ1792) 作者简介 王昊(1980—),男,教授,博士生导师 通信作者 秦严严,qinyanyan@seu.edu.cn 文章历史 收稿日期: 2017-11-14
Contents -->Abstract Full text Figures/Tables PDF
网联车混合交通流渐进稳定性解析方法
王昊1,2,3, 秦严严1,2,3
![通讯作者](http://hit.alljournals.cn/html/hitxb_cn/2019/3/images/REcor.gif)
![qinyanyan@seu.edu.cn](http://hit.alljournals.cn/html/hitxb_cn/2019/3/images/REemail.gif)
1. 城市智能交通江苏省重点实验室(东南大学), 南京 210096;
2. 现代城市交通技术江苏高校协同创新中心 (东南大学), 南京 210096;
3. 东南大学 交通学院, 南京 210096
收稿日期: 2017-11-14
基金项目: 国家自然科学基金(51478113,51878161);江苏省研究生科研与实践创新计划(KYCX17_0146);东南大学优秀博士学位论文培育基金(YBJJ1792)
作者简介: 王昊(1980—),男,教授,博士生导师
通信作者: 秦严严,qinyanyan@seu.edu.cn
摘要: 针对网联车与普通车构成的混合交通流不稳定性问题,提出一种网联车混合交通流渐进稳定性解析方法.基于传递函数理论,应用跟驰模型推导扰动在交通流中传播时的传递函数,并建立不同网联车比例下的混合交通流渐进稳定性解析框架.选取智能驾驶模型(intelligent driver model, IDM)与优化速度模型(optimal velocity model,OVM)分别作为网联车与普通车的跟驰模型,进行混合交通流渐进稳定性案例分析,并进行小扰动下的数值仿真.研究结果表明:所建立的混合交通流渐进稳定性解析框架可计算得到关于网联车比例与平衡态速度的混合交通流稳定域;当平衡态速度大于21.5 m/s时,混合交通流可在任意网联车比例下稳定,当网联车比例大于0.63时,混合交通流可在任意平衡态速度下稳定;混合交通流稳定性数值仿真实验验证了理论解析的正确性.所建立的网联车混合交通流渐进稳定性解析框架适用于不同跟驰模型的选取,能够用于分析真车实验条件下网联车对交通流稳定性的影响.
关键词: 交通流 渐进稳定性 解析研究 网联车辆 传递函数
Asymptotic stability analysis of traffic flow mixed with connected vehicles
WANG Hao1,2,3, QIN Yanyan1,2,3
![Corresponding author](http://hit.alljournals.cn/html/hitxb_cn/2019/3/images/REcor.gif)
![qinyanyan@seu.edu.cn](http://hit.alljournals.cn/html/hitxb_cn/2019/3/images/REemail.gif)
1. Jiangsu Key Laboratory of Urban ITS (Southeast University), Nanjing 210096, China;
2. Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies (Southeast University), Nanjing 210096, China;
3. School of Transportation, Southeast University, Nanjing 210096, China
Abstract: Focusing on the instability of mixed traffic flow including connected vehicles and traditional vehicles, this paper proposes an analytical method for the asymptotic stability of the mixed traffic flow with connected vehicles. Based on the transfer function theory, the car-following models were used to derive the transfer function of disturbances spreading in traffic flow. Then the analytical framework of asymptotic stability of mixed traffic flow was built under different proportions of connected vehicles. The intelligent driver model (IDM) and optimal velocity model (OVM) were selected as car-following models for connected vehicles and traditional vehicles, respectively. Subsequently, case analysis of asymptotic stability of mixed traffic flow was conducted and numerical simulations were performed under small disturbances. Results showed that the proposed analytical framework of asymptotic stability of the mixed traffic flow can be used to calculate the stability region of the mixed traffic flow related to connected vehicle proportion and equilibrium velocity. When the equilibrium velocity was larger than 21.5 m·s-1, the mixed traffic flow was stable under any connected vehicle proportion. When the connected vehicle proportion was larger than 0.63, the mixed traffic flow was stable under any equilibrium velocity. Moreover, the numerical simulation experiments of mixed traffic flow validated the soundness of the theoretical analysis. The analytical framework of asymptotic stability of mixed traffic flow can be applied to different car-following models. It can also be used to analyze impacts of connected vehicles on traffic flow stability under the condition of real experimental tests.
Keywords: traffic flow asymptotic stability analytical study connected vehicles transfer function
车联网环境下辅助驾驶系统可应用车车无线通信系统获取前方车辆行驶状态,辅助驾驶员减少驾驶失误,为有效提升交通流运营质量提供新的途径[1].交通流渐进稳定性(后文简称稳定性)表征了交通流运营的内在属性,是一种重要的交通流特性[2],理论研究与实测数据均表明,交通流稳定性是影响交通运营质量的关键内在因素,不稳定的交通流易产生时走时停的交通震荡,进而易诱发交通拥堵,因此,网联辅助驾驶车辆(简称为网联车)与普通驾驶车辆构成的混合交通流稳定性研究受到学关注[3-4].混合交通流稳定性分析往往采用数值仿真的手段进行研究,其中又分为周期性边界数值仿真实验设计[5]和开放性边界数值仿真实验设计[6].相比于单一跟驰模型稳定性理论解析,混合交通流稳定性的解析工作进展缓慢[7-9].传递函数理论[10]作为经典控制理论中的稳定性分析方法,已被应用于跟驰模型的稳定性解析[11]、智能车辆上层控制器稳定性设计[12]、以及智能车队稳定性分析中[13].但鲜有文献将其应用至网联车在不同比例下的混合交通流稳定性研究中.鉴于此,本文从网联车与普通车跟驰模型的一般性模型公式出发,应用传递函数理论,建立该混合交通流稳定性的一般性解析框架,并选择具体跟驰模型进行混合交通流稳定性案例分析.为避免混淆,本文中网联车与普通车均为小汽车车型.
1 网联车混合交通流本文中网联车是指可应用车车通信技术实时获取前车行车状态,进而辅助人工驾驶员驾驶的车辆;普通车辆是指不依靠网联辅助驾驶的传统人工驾驶车辆.不同比例下的网联车与普通车随机混在一起构成混合交通流,由于普通车辆无法提供车车通信服务,使得紧跟普通车的网联车将无法应用车车通信系统,自然在功能上退化为普通车.同时,网联车在退化为普通车之后,其车载车车通信系统仍然保留,可为后面紧跟的网联车提供车车通信服务.该退化机理与文献[14]中协同自适应巡航控制(cooperative adaptive cruise control,CACC)车辆退化为自适应巡航控制(adaptive cruise control,ACC)车辆的退化机理一致,令p表示退化前网联车的比例,则借鉴文献[14]中退化现象的数学解析表述,退化现象发生后数学期望水平下的实际网联车与普通车比例分别为
${P_1} = {p^2},$ (1)
${P_2} = 1 - {p^2}.$ (2)
其中P1为退化后的网联车比例,P2为退化后的普通车比例.
2 稳定性解析框架将跟驰模型表达为一般性的模型公式, 即
${a_n}\left( t \right) = {f_n}\left( {{v_n}\left( t \right),{h_n}\left( t \right),\Delta {v_v}\left( t \right)} \right).$ (3)
其中:fn为车辆n的模型公式,an(t)为车辆n在t时刻的加速度,vn(t)为车辆n在t时刻的速度, hn(t)为车辆n在t时刻与前车的车间距,Δvv(t)为车辆n在t时刻与前车的速度差.
应用一阶泰勒公式,对跟驰模型的一般性模型式(3)在平衡态进行线性化,得
${a_n}\left( t \right) = f_n^v\left( {{v_n}\left( t \right) - \bar v} \right) + f_n^h\left( {{h_n}\left( t \right) - \bar h} \right) + f_n^{\Delta v}\Delta {v_n}\left( t \right).$ (4)
其中fnv、fnh、fnΔv分别为跟驰模型公式在平衡态对速度、车间距和速度差的偏微分项.
定义平衡态交通流的速度扰动与车间距扰动,分别为
${u_n}\left( t \right) = {v_n}\left( t \right) - \bar v,$ (5)
${y_n}\left( t \right) = {h_n}\left( t \right) - \bar h.$ (6)
其中:uvn(t)为平衡态速度扰动,yn(t)为平衡态车间距扰动,v、h分别为平衡态速度和平衡态车间距.
将式(5)、(6)代入式(4)中,计算得到关于扰动项的微分方程为
${{\dot u}_n}\left( t \right) = f_n^v{u_n}\left( t \right) + f_n^{\Delta v}\Delta {u_n}\left( t \right) + f_n^h{y_n}\left( t \right).$ (7)
对式(7)进行拉普拉斯变换,得到扰动在交通流传播时的传递函数为
$G\left( s \right) = \frac{{f_n^{\Delta v}s + f_n^h}}{{{s^2} + \left( {f_n^{\Delta v} - f_n^v} \right)s + f_n^h}}.$ (8)
其中G(s)为速度扰动的传递函数,s为拉普拉斯域.
针对网联车和普通车,分别用G1(s)、G2(s)表示网联车扰动传递函数和普通车扰动传递函数,即
${G_1}\left( s \right) = \frac{{f_{1n}^{\Delta v}s + f_{1n}^h}}{{{s^2} + \left( {f_{1n}^{\Delta v} - f_{1n}^v} \right)s + f_{1n}^h}},$ (9)
${G_2}\left( s \right) = \frac{{f_{2n}^{\Delta v}s + f_{2n}^h}}{{{s^2} + \left( {f_{2n}^{\Delta v} - f_{2n}^v} \right)s + f_{2n}^h}}.$ (10)
其中f1n、f2n分别为网联车和普通车跟驰模型.
将式(9)、(10)分别转换至频域,有
${G_1}\left( {{\rm{j}}w} \right) = \frac{{f_{1n}^h + f_{1n}^{\Delta v}{\rm{j}}w}}{{\left( {f_{1n}^h - {w^2}} \right) + \left( {f_{1n}^{\Delta v} - f_{1n}^v} \right){\rm{j}}w}},$ (11)
${G_2}\left( {{\rm{j}}w} \right) = \frac{{f_{2n}^h + f_{2n}^{\Delta v}{\rm{j}}w}}{{\left( {f_{2n}^h - {w^2}} \right) + \left( {f_{2n}^{\Delta v} - f_{2n}^v} \right){\rm{j}}w}}.$ (12)
依据式(1)、(2)中网联车和普通车的期望数量比例关系,不同网联车比例p下的混合交通流稳定性判别条件为
${\left| {{G_1}\left( {{\rm{j}}w} \right)} \right|^{\left( {{p^2}} \right)}}{\left| {{G_2}\left( {{\rm{j}}w} \right)} \right|^{\left( {1 - {p^2}} \right)}} \le 1,\forall w \ge 0.$ (13)
其中j、w分别为频率域的虚数与频率,| ·|为传递函数频域幅值.
将式(11)、(12)代入式(13),得
$\begin{array}{*{20}{c}}{{{\left| {\frac{{f_{1n}^h + f_{1n}^{\Delta n}{\rm{j}}w}}{{\left( {f_{1n}^h - {w^2}} \right)\left( {f_{1n}^{\Delta n} - f_{1n}^v} \right){\rm{j}}w}}} \right|}^{\left( {{p^2}} \right)}} \cdot }\\{{{\left| {\frac{{f_{2n}^h + f_{2n}^{\Delta v}{\rm{j}}w}}{{\left( {f_{2n}^h - {w^2}} \right)\left( {f_{2n}^{\Delta n} - f_{2n}^v} \right){\rm{j}}w}}} \right|}^{\left( {1 - {p^2}} \right)}} \le 1,\forall w \ge 0.}\end{array}$ (14)
即当式(14)满足时,网联车混合交通流稳定,反之,不稳定.由于f1nv、f1nh、f1nΔv以及f2nv、f2nh、f2nΔv分别为网联车跟驰模型与普通车跟驰模型在平衡态对速度、车间距与速度差的偏微分项,因此在网联车与普通车跟驰模型确定的情况下,式(14)的混合交通流稳定性判别条件由网联车比例以及平衡态速度共同确定,由此可依据该判别条件计算关于网联车比例与平衡态速度的混合交通流稳定域.
3 案例分析 3.1 模型选取针对网联车跟驰模型,文献[9]认为智能驾驶模型(intelligent driver model, IDM)可较好体现网联车车载系统辅助驾驶员驾驶的特性,因此,本文应用IDM作为网联车跟驰模型,进行案例分析. IDM模型为
$ \begin{array}{*{20}{c}}{{a_n}\left( t \right) = \alpha \left[ {1 - {{\left( {\frac{{{v_n}\left( t \right)}}{{{v_0}}}} \right)}^4} - } \right.}\\{\left. {{{\left( {\frac{{{s_0} + {v_n}\left( t \right)T - \frac{{{v_n}\left( t \right)\Delta {v_n}\left( t \right)}}{{2\sqrt {\alpha \beta } }}}}{{{h_n}\left( t \right)}}} \right)}^2}} \right].}\end{array} $ (15)
其中:α为模型最大加速度,v0为最大期望速度,s0为最小停车间距,T为安全车头时距,β为舒适减速度.根据文献[9],用于网联车跟驰模型的IDM参数取值分别为v0=33.0 m·s-1,α=4.0 m/s2,s0=2.0 m,T=2.0 s,β=2.0 m·s-2.
针对普通车跟驰模型,学者们提出了众多不同类型的跟驰模型,本文选择应用较广的优化速度模型(optimal velocity model,OVM)作为普通车跟驰模型进行案例分析. OVM的模型为
${a_n}\left( t \right) = \kappa \left[ {V\left( {{h_n}\left( t \right)} \right) - {v_n}\left( t \right)} \right].$ (16)
其中κ为敏感系数,V(·)为优化速度函数,依据文献[15],优化速度函数公式为
$V\left( {{h_n}\left( t \right)} \right) = {v_0}\left[ {1 - \exp \left( { - \frac{\lambda }{{{v_0}}}\left( {{h_n}\left( t \right) - d} \right)} \right)} \right].$ (17)
其中λ为敏感系数,d为安全停车间距.依据文献[15],OVM模型参数取值分别为κ=0.700 s-1,λ=0.999 s-1,d=1.62 m.
3.2 理论解析基于网联车跟驰模型式(15)与普通车跟驰模型式(16),计算模型公式对速度、车间距、速度差项的偏微分项,分别为
$f_{1n}^v = - \frac{{4\alpha {v^3}}}{{v_0^4}} - \frac{{2\alpha T\left[ {1 - {{\left( {\frac{v}{{{v_0}}}} \right)}^4}} \right]}}{{{s_0} + vT}},$ (18)
$f_{1n}^h = 2\alpha \frac{{\left[ {1 - {{\left( {\frac{v}{{{v_0}}}} \right)}^4}} \right]\sqrt {1 - {{\left( {\frac{v}{{{v_0}}}} \right)}^4}} }}{{{s_0} + vT}},$ (19)
$f_{1n}^{\Delta v} = \sqrt {\frac{\alpha }{\beta }} \frac{{v\left[ {1 - {{\left( {\frac{v}{{{v_0}}}} \right)}^4}} \right]}}{{{s_0} + vT}},$ (20)
$f_{2n}^v = - \kappa ,$ (21)
$f_{2n}^h = \kappa \lambda \left( {1 - \frac{v}{{{v_0}}}} \right),$ (22)
$f_{2n}^{\Delta v} = 0.$ (23)
将式(18)~(23)代入网联车混合交通流稳定性的一般性判别式式(14),计算不同网联车比例、不同平衡态速度下是否满足式(14)的稳定性要求,即计算得到关于网联车比例与平衡态速度的网联车混合交通流稳定域,如图 1所示.
Fig. 1
![点击查看原图](http://hit.alljournals.cn/html/hitxb_cn/2019/3/PIC/hebgydxxb-51-3-88-1.jpg)
在图 1中,蓝色区域为混合交通流不稳定区域,绿色区域为稳定区域,图 1计算了网联车混合交通流在各网联车比例下(0~1)、以及自由流速度范围内(0~33.0 m/s)各速度取值条件下的稳定性情况.当网联车比例为0,即传统普通车交通流时,当平衡态速度小于21.5 m/s时,交通流不稳定,当速度处于21.5~33.0 m/s时,交通流稳定.在网联车比例等于1时,网联车交通流在自由流速度范围内均稳定.因此,由图 1可看出,当平衡态速度大于21.5 m/s时,混合交通流可在任意网联车比例下稳定.此外,速度越小,使得混合交通流稳定的网联车比例临界值越高,当网联车比例大于0.63时,混合交通流可在任意速度下稳定.同时,图 1能够计算得到任意速度下对应的混合交通流稳定时的网联车比例临界值,譬如,在速度为15 m/s时,若要求混合交通流能够处于稳定状态,则需网联车比例不小于0.46.
3.3 数值仿真应用3.1节案例分析中选取的网联车与普通车跟驰模型,进行混合交通流稳定性数值仿真,验证理论解析的正确性.数值仿真实验针对40辆车的车队,车队中网联车与普通车的相对数量由网联车比例随机确定,且各车辆相对空间位置亦具有随机性,并按照第1节中的描述,紧跟普通车的网联车将退化为普通车.车队以15 m/s的速度行驶于平衡态,头车产生-0.5 m/s2的小扰动打破平衡态,小扰动持续2 s,然后头车保持14 m/s的恒定速度行驶至仿真结束,仿真步长为0.1 s,数值仿真结果如图 2所示.
Fig. 2
![点击查看原图](http://hit.alljournals.cn/html/hitxb_cn/2019/3/PIC/hebgydxxb-51-3-88-2.jpg)
图 2给出了网联车比例分别为0、0.3、0.5、0.7、1时混合交通流各车辆速度随时间的变化情况.其中,黑色曲线表示普通车的速度变化情况,红色曲线表示由网联车退化为普通车的速度变化情况,蓝色曲线表示未退化的网联车速度变化情况.由图 2可以看出,网联车比例的增加可改善混合交通流稳定性,当网联车比例达到约0.5时,使得混合交通流从不稳定状态转变为稳定状态,这与图 1理论解析中在15 m/s速度下混合交通流变为稳定的网联车比例临界值为0.46基本符合.
4 结论1) 针对紧跟普通车的网联车退化为普通车的混合交通流,从跟驰模型的一般性表达式出发,基于传递函数理论,推导并建立了不同网联车比例下混合交通流稳定性的一般性解析框架.基于具体跟驰模型的案例分析表明,当速度处于21.5~33.0 m/s时,任意网联车比例下的混合交通流均稳定;当网联车比例达到0.63及以上时,混合交通流可在任意平衡态速度下稳定.
2) 建立的网联车混合交通流稳定性解析框架能够适用于网联车与普通车跟驰模型选取的多样性,能够计算关于网联车比例与平衡态速度的混合交通流稳定域.
3) 本文案例中跟驰模型的参数借鉴了国外数据标定结果,研究了网联车对混合交通流稳定性的影响.但是,仍然缺乏中国交通流实测数据的采集与标定,分析网联车对中国典型路段交通流稳定性的影响是下一步的研究工作.
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