## 2 Macro-Micro Physical Processes of Traffic Management
### Multi-Scale Mixed Traffic Modeling
 \tag{3}\]  
The motion equation for the high-order macroscopic traffic flow can be written as  
\[\frac{\text{d}u(x,t)}{\text{d}t}=\frac{\partial u(x,t)}{\partial t}+u\frac{ \partial u(x,t)}{\partial x}=\frac{u_{c}(k)-u(x,t)}{T}+\frac{\Delta x}{\tau} \frac{\partial u(x,t)}{\partial x}+F_{\text{physical}} \tag{4}\]  
\[u_{e}=V\bigg{(}\frac{1}{k}\bigg{)}=6.75+7.91 tanh(0.13/k-1.57) \tag{5}\]  
where \(v_{n}\), \(\Delta x_{n}=x_{n-1}-x_{n}\) are the nth vehicle's velocity and headway at time \(t\) (\(x_{n}\) is the \(n\)th vehicle's position), \(k\) is the reaction time, and \(V\) (\(\cdot\)) is the optimal velocity function, respectively. The conservation equation for the macro continuum traffic flow models is in the following form:  
\[\frac{\partial k}{\partial t}+\frac{\partial(ku)}{\partial x}=q \tag{6}\]  
where \(q\) is the generation rate and is set to 0 for simplification. The macroscopic traffic flow model consists of Equations (4) and (6). In the multi-scale mixed traffic model proposed in this paper, the macro traffic flow model (Equations (4) and (6)) is used to calculate driving behavior for the long time scale, while the micro car following model (Equation (1)) is used for the short time scale. Therefore, the multi-scale mixed traffic flow model is in the following form:  
Micro model of short time scale  
\[\frac{\text{d}v_{n}}{dt}=k(V(\Delta x_{n})-v_{n})+\lambda\Delta v_{n}+F_{ \text{physical}} \tag{7}\]  
Macro model of long time scale  
\[ \begin{array}{ll}\frac{\partial k}{\partial t}+\frac{\partial(ku)}{ \partial x}=q\\ \frac{\partial u(x,t)}{\partial t}+u\frac{\partial u(x,t)}{\partial x}=\frac{ V(1/k)-u(x,t)}{T}+\frac{\Delta x}{\tau}\frac{\partial u(x,t)}{\partial x}+F_{ \text{physical}}\end{array}  \tag{8}\]  
Step 2: Establishing the vehicle dynamics model with the acceleration and velocity obtained in step 1 as the boundary conditions. With this model, the feasible working point set of the engine, motor and transmission system, which determine microphysical processes, can be obtained corresponding to the driving behavior in the prediction horizon of Step 1. The structure of the PHEV vehicle used in this paper is shown in Figure 3. The vehicle in this project has a single axle hybrid system, and thus the engine and motor have the same speed. The main parameters of the vehicle are shown in Table 1.  
With the acceleration and velocity obtained in Step 1, the required torque \(T_{CVT}\) and velocity ratio \(i_{CVT}\) of CVT transmission can be obtained according to the following formula:  
\[\eta_{T}\cdot i_{CVT}\cdot i_{d}\cdot T_{CVT}+T_{b}=\frac{dv_{n}}{dt}+\left[mgf_ {r}\cos\theta+\frac{1}{2}C_{D}\cdot\rho\cdot A\cdot v_{n}^{2}+mg\sin\theta \right]\cdot\frac{1}{\delta\cdot m\cdot r} \tag{9}\]  
where \(\eta_{T}\) is the transmission efficiency; \(i_{CVT},i_{d}\) are the numerical ratios of the CVT and the final drive, respectively; \(T_{CVT},T_{b}\) are the torque of CVT and brake system; \(m\) is the vehicle weight; \(f_{r}\) is the rolling resistance coefficient; \(\theta\) is the road slope angle; \(\delta\) is the coefficient of rotational mass; \(\rho\) is the air density; \(C_{D}\) is the aerodynamic drag coefficient; \(A\) is the vehicle's frontal area; \(r\) is the wheel radius; and \(g\) is the gravitational acceleration. The pump velocity \(\mathrm{n_{HTC}}\) and pump torque \(T_{HTC}\) of a hydraulic torque converter can be obtained by \(K\)-factor and \(C\)-factor [48]:  
\[ \begin{array}{l}K_{factor}=f_{K}\left(\frac{n_{CVT}}{n_{HTC}}\cdot\frac{T_{ CVT}}{T_{HTC}}\right)\\ C_{factor}=f_{C}\left(\frac{n_{CVT}}{n_{HTC}}\cdot\frac{T_{CVT}}{T_{HTC}}\right) \end{array}  \tag{10}\]  
The torque of the hydraulic torque converter is the coupling torque of the engine and the motor:  
\[T_{HTC}=T_{\mathrm{e}}\eta_{e}+T_{m}\eta_{m} \tag{11}\]  
The engine efficiency, motor/generator efficiency and fuel consumption rate are obtained through the bench test:  
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