Network reliability-based optimal toll design

Journal of Advanced Transportation, Fall, 2008 by Hao Li, Michiel C.J. Bliemer, Piet H.L. Bovy

In reality, capacity is not a continuous variable, dependent on the number of lanes. However in order to capture the stochastic properties of link capacity caused by weather, etc, we assume link capacity as a continuous variable, following a class of distribution. Current research applies different assumptions of the distribution of stochastic link capacity, such as uniform distribution, normal distribution, exponential distribution, gamma distribution, weibull distribution (Brilon, Geistefeldt et al., 2005), etc. Depending on different characteristics of road facilities in different area, different link capacity distributions can be chosen. In our analysis, we assume each link capacity [C.sub.a] following a normal distribution independently with [omega]% coefficient of variation (value to be derived from empirical data; see AVV, 1999), expressed as:

[C.sub.a] N([[bar.C].sub.a], [([omega][[bar.C].sub.a]).sup.2])

where [[bar.C].sub.a] is the mean capacity.

The travel demand [Q.sup.rs] (k) (veh/h) for each OD pair (r,s) and departure period k [member of] K is assumed to follow a normal distribution independently as well with [beta]% coefficient of variation (derived from empirical data in Tu (2008)), expressed as:

[Q.sup.rs] (k) N([[bar.Q].sup.rs](k), [([beta][[bar.Q].sup.rs] (k)).sup.2]) (3)

where [[bar.Q].sup.rs] (k) is the mean elastic demand for time period k. It should be noted that this elastic demand is derived from some base travel demand [Q.sup.rs.sub.o] (k) and depends on the generalized OD costs [c.sup.rs] (k) ([euro]) which include the tolls denoted by the vector of (uniform) link tolls [theta] [equivalent to] [[[theta].sub.a]] ([euro]). The mean elastic demand is calculated using a demand elasticity of price, [kappa], and formulated as:

[[bar.Q].sup.rs] (k) = [Q.sub.rs] (k) (1 - [kappa] [[c.sup.rs](k, [theta]) - [c.sup.rs] (k, 0)/[c.sup.rs] (k, 0)]), (4)

where [c.sup.rs] (k, [theta]) and [c.sup.rs] (k, 0) are the generalized OD costs from r to s departing at time k with and without tolls, respectively. In this paper the generalized OD cost is calculated by the route flow weighted average route costs [c.sup.rs.sub.p] (k) for each route p [member of] [P.sup.rs]. That is,

[c.sup.rs] (k) = [[summation over (p [member of][P.sup.rs])] [q.sup.rs.sub.p] (k) [c.sup.rs.sub.p] (k)/[summation over (p [member of][P.sup.rs])] [q.sup.rs.sub.p] (k), (5)

where [q.sup.rs.sub.p] (k) is the route flow rate (veh/h) of route p from r to s departing at time k.

Before defining the route costs, we look at link cost formulation. With tolls in the network, link cost [c.sub.a] (t) of entering link a at time t is composed of the link travel time [[tau].sub.a] (t) and a link toll cost [[theta].sub.a] (t) when entering this link:

[c.sub.a] (t) = [alpha][[tau].sub.[alpha]](t) [[theta].sub.a] (t), (6)

where [alpha] ([euro]/h) is the value of time (VOT). Assuming additive link toll costs, the generalized route costs can be computed as:

[c.sup.rs.sub.p] (k) = [summation over (a [member of] p) [summation over (t)] [[delta].sup.rs.sub.ap] (k, t) [c.sub.a] (t) (7)