# Comprehensive Review of Traffic Modeling: Towards Autonomous Vehicles
## 5 Traffic Flow Models
### Macroscopic Models
#### 5.1.1 Kinematic Waves and Lane Changing
Their work was complemented by [PERSON] and [PERSON] [84], who analyzed the impact of lane changing on the relaxation phenomenon, demonstrating how these interactions influence traffic stability. Further contributions to the understanding of traffic interactions include the work by [PERSON] and [PERSON] [131], who modeled non-cooperative vehicle interactions, where slow vehicles act as moving bottlenecks for faster vehicles, leading to anisotropic behavior in the traffic stream. This model highlighted the importance of vehicle heterogeneity in traffic flow dynamics. [PERSON] and [PERSON] [92] extended this line of research by describing the spontaneous emergence of oscillations and their transformation into stop-and-go waves, linking microscopic interactions with macroscopic traffic patterns. [PERSON] [132] provided a comprehensive analysis of the bottleneck effects caused by lane changes in highway merging, diverging, and weaving areas, both theoretically and empirically. His findings underscored the significant impact of lane changing on the overall traffic flow, particularly in complex traffic environments. Recent advancements have leveraged machine learning and deep learning techniques to further refine traffic flow modeling. [PERSON] et al. [133] applied a deep learning neural network architecture to forecast a congestion score, utilizing [PERSON]'s simplified theory of kinematic waves. This approach represents a significant step towards integrating traditional traffic theories with modern data-driven methods. The work by [PERSON] et al. [134] complements this by proposing a model based on the Lagrangian discretization of the LWR model, ensuring consistency at the macroscopic scale with the propagation of congested waves according to first-order traffic flow theory. [PERSON] [135] also introduced a link queue model as a system of ordinary differential equations, which captures queue spillbacks and interactions among links. This model is a space-continuous approximation of the kinematic wave model and can be integrated into a multiscale modeling framework for network traffic flows. Meanwhile, [PERSON] et al. [136] proposed a generalized framework of the Lighthill-Whitham-Richards (LWR) model, examining vehicular flow under varying cooperative adaptive cruise control (CACC) penetration rates. They theoretically demonstrated that the kinematic wave speed propagating through the mixed platoon corresponds to the slope of the mixed fundamental diagram. Further extending the application of kinematic wave theory, [PERSON] et al. [137] proposed a dynamic Bayesian graph convolution network to infer traffic congestion propagation, allowing for simulations of congestion processes in customizable scenarios. This model represents a novel integration of traffic flow theory with advanced machine learning techniques. [PERSON] et al. [138] focused on the energy consumption aspects of traffic flow, proposing a model that reflects the characteristics of traffic kinematic waves induced by congestion, thus linking traffic dynamics with environmental impact considerations. In summary, these studies collectively demonstrate the evolution of kinematic wave models from foundational theories to modern applications that integrate behavioral dynamics, lane-changing effects, and machine learning techniques. By refining and expanding upon these models, researchers have developed a robust theoretical and practical framework that continues to inform traffic flow analysis and urban planning.