Existing traffic models are widely used in multiple frameworks, most prominently, microscopic vehicle-scale occurring on the scale of seconds and macroscopic city-scale flow patterns that develop over the scale of hours. Research works and practical applications usually employ either one or the other framework, and there is little overlap in the respective research communities. This dissertation develops mathematical techniques to bridge the two scales. The particular case of bridging the micro and macro scales of models in the stable traffic regime has been extensively studied, however what has been often overlooked is the unstable regime. Thus, of particular importance are models that can capture dynamic instabilities and traveling traffic waves called phantom jams. Such models are particularly challenging to analyze, as many papers on PDE models explicitly exclude the unstable situation. This thesis (i) outlines the mathematical foundations of microscopic and macroscopic models of interest, (ii) establishes a principled procedure of generating macroscopic flow quantities from microscopic models in the unstable regime, (iii) presents a study addressing the averaging of scales and the understanding of macroscopic manifestations of microscopic car-following traffic waves based on a framework of systematic hierarchy of tests that isolate the car-following dynamics, (iv) explains the corresponding effective traffic state and non-equilibrium wave structures that rise in the fundamental diagram, (v) and derives and validates vehicle type specific simple fuel consumption rate models that are accurate, computationally fast, and have desirable physics-like properties. The insights gained from this study has many applications. One of them presented here is the relevance of dampening traffic waves in the presence of sparse control and in light of the energy demand of traffic at the vehicle-scale, waves-scale, and city scale. / Mathematics

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