Outline. 3. Implementation. 1. Introduction. 2. Algorithm

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2 Outline 1. Introduction 2. Algorithm 3. Implementation

3 What s Dynamic Traffic Assignment? Dynamic traffic assignment is aimed at allocating traffic flow to every path and making their travel time minimized over the time. Dynamic traffic assignment belongs to traffic planning, it plays an important role in Intelligent Transportation System Such as Route Guidance in Google map Heat Map in Baidu map

4 Dynamic traffic assignment is the positive modeling of time-varying flows of automobiles on road network consistent with established traffic flow theory and travel demand theory.

5 Continuous Time Dynamic User Equilibrium (DUE) Desired solution:

6 Continuous Time Dynamic User Equilibrium (DUE) For each individual, compared with your current travel cost: Until the Nash equilibrium is reached!

7 Nash equilibrium In game theory, the Nash equilibrium, named after American mathematician John Forbes Nash Jr., is a solution concept of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

8 Solution? How s the transportation system going to look like under that equilibrium? Fig Departure rates and corresponding travel cost in the DUE solution

9 Solution? To know the equilibrium strategies of the other players: Dynamic Network Loading: The problem of finding link activity when travel demand and departure rates (path flows) are known is commonly referred to as the dynamic network loading problem. To find the equilibrium: Differential Variational Inequality (dvi)

10 Progress dx ai dx a1 p (t൯ = h p (t) g p a1 (t) p P p dt (t൯ p = g dt ai 1 (t) g p ai (t) p P, i [2, num(p) dg p ai (t) dt dr ai = r p ai (t) p P, i [1, num p ] dr p a1 (t) dt = R p a1 (x, g, r, h) p P p (t൯ = R p dt ai (x, g, r) p P, i [2, num(p) x p ai ((τ 1) Δ) = x p,0 ai p P, i [1, num(p) ൧ g p ai ((τ 1) Δ) = 0 r p ai ((τ 1) Δ) = 0 p P, i [1, num(p) ൧ p P, i [1, num(p) ൧

11 Arc Volume (x) Arc Delay Path Delay Travel Cost

12 Progress DVI(Ψ,Λ, [t 0,t f ]): find h Λ 0 such that p P න t 0 t fψp (t, h )(h h )dt 0 h Λ where Λ = h 0 : dy ij dt = p P ij h p (t), y ij (0) = 0, y ij (t f ) = Q ij

13 Progress h = P Λ [h αψ p (t, h )൧ t f න hp k (t) αψ(t, h k p ) + v ij t 0 p P ij + = Q ij h p k+1 = h p k (t) αψ(t, h p k ) + v ij +

14 Algorithm 1 Computing ODE and fixed point iteration Flow chart Initialization: path, timespan, arc, h0, Q(demand), epsilon (tolerance) et. while condition is true solve ODE to get link volume get link delay get effective path delay solve v update hk+1 end while Output: Phi, hk hk - hk+1 is larger than tolerance ODE Dynamic Network Loading Get arc volume Get arc delay Get path delay DVI Problem The best departure time and path choice Fixed-Point Problem

15 Arc Jam density (vehicles/km) Free flow speed (km/5min) Length (km)

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30 epsilon = 0.05 Using openmp Without using openmp

31 siouxfall 23 pairs siouxfall 10 pairs small Using openmp Without using openmp

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36 ρ(t, x) t + f(ρ(t, x)) x = 0

37 Flow Capacity Free Congestion Da t = q in a (t L a k a ) C a if N a up (t L a a ) = N k down (t) a ifn a up (t L a a ) > N k down (t) a Demand Capacity ` density Free Congestion S a (t) = q a out (t L a ) w a C a if N a a up (t) = N down (t L a a ) + ρ w jam a if N a a up (t) < N down (t L a a ) + ρ w jam a L a L a Supply Capacity Free Congestion density density

38 α J ij (t) = μ p a (t, L a ൯ p a,b S j (t൯ q out,i = min{d i (t), } j I o α i q in,j = α ij q out,i (t൯ i I v P1 P2 P3 a N down (t) = N a up (τ a (t) ൯ travel_time = τ a (t) t

39 Algorithm 2: Computing dynamic network loading based on LWR model by C Flow chart Initialization: path,timespan, arc, h0, Q(demand), epsilon (tolerance) et. for all i = 1 : num (OD pair) While condition is true hk - hk+1 is larger than tolerance for t =1 : num (timesteps) for i = 1 : num (links) Solve D Get link demand equation 5.2 Solve S Get link supply equation 5.3 for j = 1 : num (linkin) for k = 1 : num (linkout) get turning ratio equation 5.4 end for end for end for Calculate entering flow equation 5.5 Calculate exiting flow equation 5.6 end for get effective path delay get a v in each iteration equation 3.3 update hk+1 according to equation 3.4 each v map to a hk end while end for Output: Phi, hk Input T<time steps Get link demand suply Get junction turning ratio Update entering flow Update exiting flow Output The best departure time and path choice Fixed point iteration DVI problem Get path delay Get link delay

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45 Thank You