# A Hybrid Control Path Planning Architecture Based on Traffic Equilibrium Assignment for Emergency
## 3 Preliminaries  
### Dynamic Road Network and Traffic Equilibrium Principle  
A generalized dynamic road network \(G=(N,R,O^{T},V^{T})\) is obtained by combining the road network topology and the travelers' trajectories. The OD evacuation volume \(O^{T}\) and background load \(V^{T}\) change dynamically with the period \(T\). The origin and destination of the OD come from two nodes in set \(N\). The set of road sections in the network is denoted by \(R\). We refer to the connection from the origin to the destination as the OD path \(l\). The \(l\) consists of a series of adjacent link sequences \(<\!\!r_{o},r_{i},\ldots,r_{d}\!\!>\) which correspond to the road sections in the road network. Meanwhile, as the traffic load changes, the impedance of the road section varies simultaneously, and we use an evacuation time function to measure it.  
Definition 1 (Traffic Load): The total traffic volume carried by the road network during the time period, including background traffic flow loads (background load) as well as assigned evacuation volumes. The total traffic volume allocated to each section is called the traffic load of that section, and the maximum traffic load is equal to its capacity. Therefore, the real-time traffic load combined with the section capacity can be used to calculate the actual capacity.  
Definition 2 (Travel Impedance): For the edge \(r_{i}\) in the OD path, the \(\delta(r_{i})\) denotes the evacuation time cost function. The travel impedance for \(r_{i}\) can be expressed in Equation (1).  
\[C(r_{i})=\{\delta(r_{i})|<r_{i}>\ \in l\}. \tag{1}\]  
_The travel impedance of path \(l\) is the total cost of the containing road sections, as in Equation (2):_  
\[C(l){=}\delta^{(t_{0})}(r_{o})+\delta^{(t_{1})}(r_{i}){+}\cdots{+}\delta^{(t_{k })}(r_{d}). \tag{2}\]  
_where \(t_{0},t_{1},\ldots,t_{k}\) denote the discretized time points, and \(k\) means the time steps spent on \(l\)._  
In the evacuation of ODs, the travel impedance of the pre-planned path may increase due to the traffic load changes. The increasing traffic load on the path causes the pursuit of minimum evacuation costs \(C(l)\) to be invalidated, while some sections with low flow traffic are not fully utilized. The problem domain for hybrid control path planning is based on the criteria that the total evacuation cost of all ODs in the traffic network is minimized. The principle of system optimization for traffic assignment is used here.  
**Definition 3** (System Optimization, SO [2]): _With the constraint of traffic load balancing, traffic flows in a congested network should be assigned at the lowest average or total evacuation cost._  
The SO requires travelers to accept the system's unified scheduling, where there is a collaborative relationship between them. Ref. [3] constructed a mathematical model to achieve system optimization under constraints, and regarded the OD flow as an equilibrium distribution of the network. This objective optimization function is shown in Equation (3).  
\[\begin{array}{l}\min z(L,V_{L})=\sum\limits_{r_{i}\in L,\frac{r_{i}}{r_{i}} \in V_{L}}v_{r_{i}}C(r_{i})\\ s.t.\quad V_{L}{=}\sum\limits_{l\in L}\sum\limits_{r_{i}\in l}v_{r_{i}}^{l};v_{ r_{i}}^{l}\geq 0\end{array}. \tag{3}\]  
The \(z(\cdot)\) is the sum of the product of travel impedance and traffic load, and its optimal solution minimizes the function value. The constraints include the conservation of traffic between OD paths and road sections with non-negative section traffic. The \(V_{L}\) denotes the traffic volume of all OD paths \(L\), and \(v_{r_{i}}^{l}\) denotes the traffic load of \(r_{i}\) in \(l\). As the Beckmann-based traffic assignment methods use stepwise approximation, they are difficult to apply in large-scale traffic networks with high real-time requirements [43]. In this paper, we design and map the equilibrium traffic assignment method to data-driven deep learning, which is based on SO ideas for online real-time decision-making in large-scale transportation networks.