8. Macroscopic Traffic Modeling
Introduction In traffic stream characteristics chapter we learned that the fundamental relation (q=k.u) and the fundamental diagrams enable us to describe the traffic state of stationary and homogeneous traffic. Thus we can calculate the two remaining variables for a given value of a macroscopic variable.
Introduction When traffic is stationary and homogeneous, we know that the values for these variables will remain constant along the entire road and for some extended period. However, real traffic is neither homogeneous nor stationary
Introduction In doing so, we define the dynamic relation between q(x,t), u(x,t) and k(x,t). We assume, therefore, that we are dealing with point variables: variables that are singularly defined at any moment and at every location. By doing this we can show these three variables as functions in the t-x plane.
Macroscopic Traffic Flow Modeling Tries to describe the aggregate behaviour of the flow characteristics Various analogies use to describe traffic flow Fluid flow analogy Heat flow analogy Gas flow analogy
Macroscopic Traffic Flow Modeling Hydrodynamic analogy Law of conservation of number of vehicles Balance continuity (conservation) equation k( x, t) t + q( x, t) x = 0 k and q are unknowns.
Derivation of conservation equation The conservation equation can easily be derived by considering a unidirectional continuous road section with two counting Stations 1 and 2 (upstream and downstream, respectively)
Derivation of conservation equation Let N be the number of cars (volume) passing Station i during time t and q i, the flow passing station i; t is the duration of simultaneous counting at Station 1 and 2. Without loss of generality, suppose that N 1 >N 2. Because there is no loss of cars x (i.e., no sink), this assumption implies that there is a buildup of cars between Station 1 and Station 2.
Derivation of conservation equation Let (N 2 N 1 ) = N; for a buildup N will be negative. Based on these definitions we have then the build-up of cars between stations during t will be (- q) t: D C B A Situation at time t 1
Derivation of conservation equation Let (N 2 N 1 ) = N; for a buildup N will be negative. Based on these definitions we have then the build-up of cars between stations during t will be (- q) t: D C B A Situation at time t 2
Derivation of conservation equation D C B A
Derivation of conservation equation D C B A
Derivation of conservation equation
Solution of the conservation equation The conservation equation is a state equation that can be used to determine the flow at any section of the roadway. The attractiveness of this equation is that it relates two fundamental dependent variables, density and flow rate, with the two independent ones (i.e., time t, and space x). Solution of the equation is impossible without an additional equation or assumption. k ( x, t) t + q( k ( x, t)) x = 0
Solution of the conservation equation Lighthill and Whitham solved the equation by assuming flow rate q is a function of local density k. k(x,t) and q(x,t) are dependent, so q(k(x,t)) Therefore the continuity equation takes the form: k ( x, t) t + q( k ( x, t)) x = 0
Daganzo simplification of the solution Daganzo further simplified the solution scheme of Lighthill and Whitham by adopting the following relationship between q and k: q { v k Q wk k } = min,, ( ) f jam Daganzo named his model as: Cell Transmission Model (CTM)
Cell Transmission Model (CTM) Time is discretized into equal intervals of t (1, 2, 3, 5s) Networ k is divided into segments called cells Cell properties Cell length = x = v f. t n i N i q i Q i : Cell occupancy (actual number of vehicles) : Maximum possible cell occupancy : Actual inflow : Inflow capacity i-1 i i+1 n i n i+1 n i-1 q i q i+1
Cell Transmission Model (CTM) Approximation of q-k relationship q Q V f k w k v f -w q = min { v f. k ; Q ; w. (k max k) } With k = n/ x = n/(v f * t) for t = 1 => i-1 i i+1 n i-1 n i n i+1 q i q i+1 x x x k max k Waiting vehicles q i = min { n i-1 ; Q i ; w/v f. (N i n i ) } Inflow capacity Available space
Cell Transmission Model (CTM) Continuity equation q = k x t t t t+ 1 t qi+ 1 qi = ni n i i-1 i i+1 n i-1 n i n i+1 q i q i+1 x x x t+1 n i = t n i t +q i t qi + 1
Simulation of Traffic Signal i-1 i i+1 n i n i+1 q i q i+1 n i-1 For t red phase For t green phase q(t) i = 0 As previous equation q i = min { n i-1 ; Q ; w/v f. (N i n i ) }
d I (t) Derivation of Delay Equation By Almasri 2006 t d(t) i = t v(t) i vf t where: q(t) k v(t) I = k(t) t v f n i q k v f v I (t) For k(t) v I f i n(t) i = x x = t For one vehicle n(t) i [ ] d(t) = t n(t) q (t) t I i k [ ] d(t) = n(t) q (t) t=1 I I k No. of vehicles Once delay is determined at cell level, it can be determined at link or network level, by summing up the delays for all cells.
Cell Representation and Boundary Conditions When the free flow speed is 50 kph and the simulation 28m step is 1s, then the length of each cell in the network is 13.89m ((50/3.6) 1). 42 m 28m Therefore, the input and exit sections should have 3 and 2 cells respectively 42 m
Cell Representation and Boundary Conditions 16 15 14 1 2 3 4 5 6 7 8 13 Origin Cell Gate Cell 12 Destination Cell 11 Normal Cell 10 9
Traffic Management and Control (ENGC 6340) Cell Representation and Boundary Conditions Origin, destination, and gate cells are used to specify boundary conditions as follows: Origin cells (e.g. cells 1and 9) must have an infinite number of vehicles (n =infinity ) that discharge into empty gate cells. 16 15 Gate cells (e.g. cells 2 and 10 in) must have an infinite size (N =infinity) and the inflow capacities of the cells Q(t) are set equal to the desired section input flow for time interval t. 1 14 2 3 4 5 6 7 13 Origin Cell 12 8 Destination cells (e.g. cells 8 and 16), where traffic flows terminate and exit the network, should have infinite sizes (N = infinity). Gate Cell Destination Cell Normal Cell 11 10 9
Traffic Management and Control (ENGC 6340) Calculation Procedure 1. Initialize the cell occupancy (n) based on the input data of initial occupancy proportion (n/n) either with zeros when the network is empty or with definite values when the network is preloaded. 2. Calculate the flows q(t) for the first time step in all cells using the already described flow equations without any restriction of cell order. q i = min { n i-1 ; Q i ; w/v f. (N i n i ) } 3. Calculate the delays d(t) for the first time step in all cells using the already derived delay equations also without any restriction of cell order. For d(t) = n(t) q (t) [ ] t=1 I I k 4. Calculate the cell occupancies n(t+1) for the next time steps in all cells. t+1 t t t n i = n i + q i qi + 1 5. Repeat steps 2-4 for each time step till the end of the time horizon.
Traffic Management and Control (ENGC 6340) Pseudo-code for the calculation steps // Initialization Initialize n I for all cells with 0 or percentages of N I // Loop For time t = 0 to the time horizon For cell I = 1 to number of cells Calculate flow q(t) at time t for all cells ( No order of calculation) End loop For cell I = 1 to number of cells Calculate delays d(t) for all cells ( No order of calculation) End loop For cell I = 1 to number of cells Calculate number of vehicles n(t+1) for all cells ( No order of calculation) End loop End loop