A Physics-Inspired Distributed Energy Equation for Macroscopic Traffic Flow Models

This paper presents a distributed, physics-based energy equation to be used with macroscopic traffic flow models. The proposed macroscopic energy equation is derived starting from the road load equation and it is then integrated with the Aw-Rascle-Zhang model. Results are compared against the energy output of microscopic car-following models, namely the Improved Intelligent Driver Model and the Extended Intelligent Driver Model. To enable the comparison between macroscopic and microscopic model variables, kernel density estimation is used. This allows individual vehicle positions, speeds, and energy usage obtained from the ordinary differential equations to be converted to distributed quantities. The proposed macroscopic energy equation is evaluated on three different traffic scenarios: free flow, congested traffic, and mixed traffic. It is shown that the proposed energy model can calculate the energy output within ±4% in free flow traffic, ±11% in congested traffic, and ±8% in mixed traffic.