  The literature about tradable credit schemes (TCS) as a demand management
system alleviating congestion flourished in the past decade. Most proposed
formulations are based on static models and thus do not account for the
congestion dynamics. This paper considers elastic demand and implements a TCS
to foster modal shift by restricting the number of cars allowed in the network
over the day. A trip-based Macroscopic Fundamental Diagram (MFD) model
represents the traffic dynamics at the whole urban scale. We assume the users
have different OD pairs and choose between driving their car or riding the
transit following a logit model. We aim to compute the modal shares and credit
price at equilibrium under TCS. The travel times are linearized with respect to
the modal shares to improve the convergence. We then present a method to find
the credit charge minimizing the total travel time alone or combined with the
carbon emission. The proposed methodology is illustrated with a typical demand
profile from 7:00 to 10:00 for Lyon Metropolis. We show that traffic dynamics
and trip heterogeneity matter when deriving the modal equilibrium under a TCS.
A method is described to compute the linearization of the travel times and
compared against a classical descend method (MSA). The proposed linearization
is a promising tool to circumvent the complexity of the implicit formulation of
the trip-based MFD. Under an optimized TCS, the total travel time decreases by
17% and the carbon emission by 45% by increasing the PT share by 24 points.
