lecture11

lecture11 - 1.225J (ESD 205) Transportation Flow Systems...

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Lecture 11 Lecture 11 Traffic Flow Models, and Traffic Flow Models, and Traffic Flow Management in Road Networks Traffic Flow Management in Road Networks Prof. Prof. Ismail Chabini Ismail Chabini and Prof. and Prof. Amedeo Amedeo R. R. Odoni Odoni 1.225 1.225 J (ESD 205) Transportation Flow Systems 1.225, 11/28/02 Lecture 9, Page 2 Lecture 11 Outline Lecture 11 Outline ± Overview of some traffic flow models: • Modeling of single link: Car-following models • Dynamic macroscopic models of highway traffic ± Dynamic traffic flow management in road networks: • Concepts • Dynamic traffic assignment • Combined dynamic traffic signal control-assignment ± The ACTS Group 1
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1.225, 11/28/02 Lecture 9, Page 3 Link Travel Time Models: Car Link Travel Time Models: Car -Following Models Following Models ± Notation: Flow n Leader n+1 Follower ± jam n n n k t l t x t x 1 ) ( headway) (space spacing ) ( ) ( 1 1 + = = + + x n+1 x n x 0 L ) ( ) ( ) ( 1 1 t l t x t x n n n + + = ± ) ( ) ( : n vehicle of speed t x dt t dx n n = ± ) ( ) ( ) ( : n vehicle of tion) (deccelera on accelerati 2 t x dt t x d dt t x d n n n = = ± ± car-following regime: l n+1 ( t ) is below a certain threshold l n+1 jam k 1 1.225, 11/28/02 Lecture 9, Page 4 Link Travel Time Models: Car Link Travel Time Models: Car -Following Models Following Models ± Simple car-following model: )) ( ) ( ( ) ( ) ( 1 1 1 t x t x a t l a T t x n n n n + + + = = + sec) 5 . 1 ( : T time reaction T ) 37 . 0 ( : 1 s a factor y sensitivit a Flow n Leader n+1 Follower x 0 x n+1 x n L ± Questions about this simple car-following model: • Is it realistic? • Does it have a relationship with macroscopic models? l n+1 jam k 1 2
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Lecture 9, Page 5 From Microscopic Models To Macroscopic Models )) ( ) ( ( ) ( 1 1 y x y x a y x n n n + + = dy t l a dy y x y x a dy y x n n n n ) ( )) ( ) ( ( ) ( 1 1 1 + + + = = + + = t n t n dy t l a dy y x 0 1 0 1 ) ( ) ( )) 0 ( ) ( ( ) 0 ( ) ( 1 1 1 1 + + + + = n n n n l t l a u t u ± Fundamental diagram: ± Simple car-following model: ± Proof of “equivalency” ) 0 ( )) ( ) ( ( ) ( 1 1 = = + + T t x t x a t x n n n ) 1 ( max jam k k q q = ) 0 ( ) 0 ( ) ( ) ( 1 1 1 1 + + + + + = n n n n l a u t l a t u 0 ) 0 ( ) 0 ( 0 ) ( then , 0 ) ( If 1 1 1 1 = = = + + + + n n n n l a u t u t l 1.225, 11/28/02 Lecture 9, Page 6 ) 1 1 ( jam k k a u = ) 1 ( ) 1 1 ( jam jam k k a k k k a k u q = = = ) 1 ( max jam k k q q = a q k = = then 0, If ) 1 ) ( 1 ( ) ( ) ( 1 1 1 jam n n n k t k a t l a t u = = + + + max then ), 1 ( Since q a k k a a q jam = = ± Note: if k 0, then u →∝ . Does this make sense? 3
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lecture11 - 1.225J (ESD 205) Transportation Flow Systems...

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