An implicit solution scheme for the Link Transmission Model

Dynamic network loading (DNL) models are at the core of a wide variety of optimization schemes and network analysis tools. In practice this calls for fast and efficient methods to calculate traffic states for various levels of accuracy and numerous adaptations to the boundary conditions. In this paper, we describe the Implicit Link Transmission Model (I-LTM) a dynamic network loading algorithm that avoids small update steps and is able to calculate adaptations of an existing solution efficiently. Within each update step, an implicit consistency problem between flow propagation and network constraints is formulated, resulting in a fixed point solution with appropriate network delays. In an iterative scheme, this consistency problem is solved using the constraints of a previous iteration. The algorithm is further optimized by limiting calculations to the part of the network that has changed. I-LTM allows for fast sensitivity analyses, optimization algorithms and calibration methods and it avoids numerical instabilities related to large time steps, typically observed in most DNL algorithms. This makes it beneficial in terms of calculation effort and robustness of the result.