# Modeling of Traffic Flows Sustainability on Highway Network Stretches
## 1 Introduction
### Hybrid Models of the Traffic Flows  
The third class of models is hybrid models of traffic flows. Hybrid models are devoted to papers [44; 45; 46; 47; 48; 49; 50]. These models combine macro and micro models' capabilities. The studies [44; 45] provide hybrid models analysis of traffic flows over the past decade. The component modules of models and architectural solutions were analyzed. Such models are designed based on neural networks [45] with self-learning and self-control functions. A hybrid model is proposed to investigate the stopping and driving phenomenon caused by passing vehicles in front, combining capabilities at macro and micro levels [46]. First, under the stochastic nature of driving behavior, Brownian noise is added to the speed difference to modify the model of vehicle passing. The traffic flow macroscopic characteristics are then added, and the two models are combined in the context of intensive transport flow. The basis for designing such models is traffic density estimation [47]. According to the authors, vehicle recognition and counting are two main steps for evaluating traffic density. Vehicle identification systems can be designed based on neural networks. In [48; 49; 50; 51; 52], hybrid models are presented as intelligent traffic control systems. These system types effectively solve traffic jam issues in urban areas, increasing road capacity and ensuring driver safety. Intelligent traffic control systems' short-term forecasting of traffic flows is the main task. After reviewing publications on hybrid models of traffic flows, it is important to mention that these models are primarily developed using neural networks. They are utilized for simulating road situations on both small and large stretches of highway networks. The main factors in such models are vehicle density and speed in traffic, fluctuations in speed (acceleration, deceleration), the driver's response to a change in the road situation, and the transport flow sustainability to traffic jams. We believe traffic flow density gradient and velocity gradient should be significant factors when using hybrid models. In the study [18], the velocity gradient of the transport flow was utilized, and in the paper [6], the alteration in flow density was analyzed. These parameters will make it possible to consider transport flow stochastic, which is noted in the investigation [21], and increase the prediction rate of the transport flow sustainability to traffic jams. The authors used the same approach in the investigation [53]. This paper proposes a set of decision-making systems for optimizing transport flow to solve traffic jam issues in a specific area. To analyze actual driving conditions and calculate passing density and speed, a driving prediction model is designed and updated iteratively. Based on this model, the congestion rate on the current highway stretch is estimated, and therefore, methods of intelligent decision-making and consistent optimization are proposed. In addition, this article implements some application experiments on a controlled highway section with three intersections and provides perfect results for predicting density and speed. Besides, the presented model in this paper has higher accuracy, shorter prediction time, and stronger interference protection than other existing prediction methods. However, this study does not answer determining the sustainability limit on congestion occurrence. In the paper [54], graph theory is used, according to relationships accounts between the spatial-local state of neighboring roads and the controlled one. In this case, traffic flow speed is predicted by delays. The model does not consider the driver's behavior and, therefore, their impact on traffic flow sustainability. The study [55] proposed a criterion for estimating traffic based on robust vector equilibrium. To estimate the current rating of traffic flow sustainability, a subspecies of the minimax task is solved. Unfortunately, the application of this method to the real road network has not been shown.