# A Hybrid Control Path Planning Architecture Based on Traffic Equilibrium Assignment for Emergency
## 3 Preliminaries
### Formulation of Hybrid Control Path Planning Problem  
Hybrid control path planning is a path decision model that combines network-wide traffic flow information (macroscopic information) and travelers' real-time OD information (microscopic information) to consider traffic system optimization. In the path-planning process, it considers both individual paths cost \(C(l)\) and system cost \(z(L,V_{L})\). This is achieved by mapping the traffic equilibrium assignment method based on SO. However, background traffic flows at the macro level are not available in real time. Traffic assignment is only based on the macro background traffic flow, which cannot adapt to the situations wherein individuals change the established route to adapt to time-varying transportation system states in emergencies. Thus, the hybrid control path planning relies on the data-driven deep learning approach to perceive future traffic evolution.  
We use the discrete time points \((t-m,...,t-1,t)\) to record the OD volume and background load of road sections. The prediction of the dynamic traffic load can be considered as learning the mapping function \(f(\cdot)\) based on \(O^{T}\) and \(V^{T}\), as shown in Equation (4).  
\[(o_{r},v_{r})^{t}=f\bigg{\{}\vec{A}_{i}\bigg{[}\begin{array}{ccc}o_{r}^{t-m },&\cdots,&o_{r}^{t-1}\\ v_{r}^{t-m},&\cdots,&v_{r}^{t-1}\end{array}\bigg{]}\bigg{\}}. \tag{4}\]The \(o_{r}^{t}\in O^{T}\) denotes the OD volume and \(v_{r}^{t}\in V^{T}\) denotes the traffic load of node \(r\) at time \(t\). The \(\bar{A}\) is the spatial connectivity of the nodes in the road network. The historical data of \(o_{r}\) and \(v_{r}\) from \(t-m\) to \(t-1\) can be represented as \((o_{r}^{t-m},\cdots,o_{r}^{t-1})\) and \((v_{r}^{t-m},\cdots,v_{r}^{t-1})\). Considering that acquiring OD data is limited for some nodes, the historical data for \(o_{r}\) can be taken from other nodes with similar spatial patterns [44]. The traffic equilibrium graph can be considered as a priori traffic distribution of \(O^{T},V^{T}\). The traffic distribution \(F\) in period \(T\) has the following decomposition equation according to the conservation of traffic flow:  
\[E(O^{T},V^{T})=(F_{o}^{T}|_{\min z(\cdot)})+F_{v}^{T}. \tag{5}\]  
Equation (5) learns an equilibrium distribution function \(E(\cdot)\) based on the SO-constrained traffic equilibrium assignment method that assigns OD evacuation volume to road sections. The hybrid control path planning builds upon the equilibrium distribution \(E_{(o,p)}^{T}\), which equals finding a path with the largest potential traffic assignment. On this basis, the system recommends that path planning prioritizes using the road sections with potential traffic, which can reach a compromise between individual cost and system optimization, as shown in Equation (6):  
\[I_{(r_{o},r_{d})}^{\prime}=HRoute(r_{o},r_{d}|r_{i}\in E_{(o,p)}^{T}). \tag{6}\]  
where \(l^{\prime}\) is the decision path under hybrid control. To capture the impact of dynamic traffic network for path planning, this paper attempts to learn a \(HRoute(\cdot)\) function that considers the overall traffic evolution trend of the system to the path planning of the individual traveler. In fact, the potential traffic assignment can better represent the evolution of the traffic network. Therefore, the hybrid control path planning model extracts the evolution features of the traffic network to calculate the individual paths.