Most time-dependent vehicle routing problems are based on a similar modeling paradigm: travel time information is represented by travel time functions between pairs of points of interest (e.g., depot or customers). Only a few papers investigate how these functions can be computed using the available travel time information. Furthermore, most of them neglect the possibility that different paths could be selected in the road network depending on the compromises they offer between cost (distance) and travel time. In this paper, we propose a new setting where travel time functions are defined on road-network arcs. We consider the Time-Dependent Vehicle Routing Problem with Time Windows and solve it with a branch-and-price algorithm. As far as we know, this is the first exact approach for a time-dependent vehicle routing problem when travel time functions are initially defined on the segments of a road-network.