基于变间距动态规划的中高速磁悬浮列车速度曲线优化
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摘要
本文首先对中高速磁悬浮列车动力学模型进行分析,之后在考虑辅助停车区对列车运行影响的基础上提出了一种变间距的动态规划模型。该模型以双速度防护曲线临界点作为划分分区的依据,之后将分区再次划分为不同间距长度的子阶段,并构建阶段内能耗以及时间变化方程,最后在满足相关约束条件下采用逆推法求解变间距的动态规划模型。本文通过仿真案例,对比了变间距动态规划模型与等间距动态规划模型在列车速度曲线优化的异同,验证了本文提出的模型在时间效率以及能耗优化上的优势。
A significant approach to save energy of the middle-to-high speed maglev is to optimize the trajectory planning. This paper establishes a model based on dynamic programming with mutative spacing after analyzing the kinetic equations of the middle-to-high speed maglev and the effect of the assistant-stop area. The model firstly uses the crossover point of double protected curve as the foundation of partition.Then, model further divides the partition into small sub-stages with different spacing and formulates the equation of energy and time respectively. Finally, the model is solved by reversing push method under certain constraints. The established model is compared the conventional dynamic programming method through the numerical examples. The results indicate the advantage of the proposed model in energy saving and computational efficiency.
A significant approach to save energy of the middle-to-high speed maglev is to optimize the trajectory planning. This paper establishes a model based on dynamic programming with mutative spacing after analyzing the kinetic equations of the middle-to-high speed maglev and the effect of the assistant-stop area. The model firstly uses the crossover point of double protected curve as the foundation of partition.Then, model further divides the partition into small sub-stages with different spacing and formulates the equation of energy and time respectively. Finally, the model is solved by reversing push method under certain constraints. The established model is compared the conventional dynamic programming method through the numerical examples. The results indicate the advantage of the proposed model in energy saving and computational efficiency.
引文
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[13]江亚,吴汶麒,刘进.磁悬浮列车运行控制系统二维速度防护曲线仿真[J].同济大学学报:自然科学版,2004,32(3):397-400.Jiang Ya,Wu Wen-lin,Liu Jin,et al.Simulation of high-speed maglev train 2-D speed protection curve[J].Journal of Tongji University(Natural Science),2004,32(3):397-400.
[14]杨光,唐祯敏.高速磁悬浮列车的安全速度防护问题研究[J].北京交通大学学报,2007,31(2):38-42.Yang Guang,Tang Zhen-min.Study on safety speed protection of the high maglev train[J].Journal of Beijing Jiaotong University,2007,31(2):38-42.
[15]Howard R A.Dynamic programming[J].Management Science,1966,12(5):317-348.
[2]杨光.高速磁悬浮列车最优速度曲线及其跟踪控制研究[D].北京:北京交通大学电子信息工程学院,2007.Yang Guang.Study on high speed maglev train optimum speed curve and its tracking control[D].Beijing:College of Electronic Information Engineering,Beijing Jiaotong University,2007.
[3]杨轲.磁悬浮列车仿真平台设计及运控算法研究[D].杭州:浙江大学电气工程学院,2008.Yang Ke.Research on maglev train simulation platform design and operation and control algorithm[D].Hanzhou:College of Electrical Engineering,Zhejiang University,2008.
[4]Ichikawa K.Application of optimization theory for bounded state variable problems to the operation of a train[J].Bulletin of Japanese Society of Mechanical Engineering,1968,11(47):857-865.
[5]Milroy I P.Aspects of automatic train control[D].Loughborough:Loughborough University,1980.
[6]Asnis I A,Dmitruk A V,Osmolovskii N P.Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle[J].USSR Computational Mathematics and Mathematical Physics,1985,25(6):37-44.
[7]Khmelnitsky E.On an optimal control problem of train operation[J].IEEE Transactions on Automatic Control,2000,45(7):1257-1266.
[8]Howlett P.The optimal control of a train[J].Annals of Operations Research,2000,98(1-4):65-87.
[9]Dong H,Zhang L,Chen Y,et al.Multi-objective train trajectory design based on dynamic programming[C]∥The 33rd Chinese Control Conference(CCC),Nanjing,China,2014:9060-9065.
[10]Xu Y,Zhao X,Wang L,et al.Optimal control of automatic train operation based on multi-scale dynamic programming[C]∥The 33rd Chinese Control Conference(CCC),Nanjing,China,2014:3429-3433.
[11]Lu S,Hillmansen S,Ho T K,et al.Single-train trajectory optimization[J].IEEE Transactions on Intelligent Transportation Systems,2013,14(2):743-750.
[12]刘进,吴汶麒.高速磁悬浮交通二维速度防护曲线及其算法研究[J].中国铁道科学,2002,23(4):106-110.Liu Jin,Wu Wen-lin.Research on 2-D speed protection curve and its algorithm of high-speed maglev transportation[J].China Railway Science,2002,23(4):106-110.
[13]江亚,吴汶麒,刘进.磁悬浮列车运行控制系统二维速度防护曲线仿真[J].同济大学学报:自然科学版,2004,32(3):397-400.Jiang Ya,Wu Wen-lin,Liu Jin,et al.Simulation of high-speed maglev train 2-D speed protection curve[J].Journal of Tongji University(Natural Science),2004,32(3):397-400.
[14]杨光,唐祯敏.高速磁悬浮列车的安全速度防护问题研究[J].北京交通大学学报,2007,31(2):38-42.Yang Guang,Tang Zhen-min.Study on safety speed protection of the high maglev train[J].Journal of Beijing Jiaotong University,2007,31(2):38-42.
[15]Howard R A.Dynamic programming[J].Management Science,1966,12(5):317-348.