# Collaborative Trajectories Optimization for Tram and Connected Vehicles at Signalized Intersection
## 3 Mathematical Model  
### Dynamic Model of Trans  
According to the operating characteristics of trans, the traction and braking forces have different maximum values at different operating speeds. The formulas for traction and braking characteristics of trans are defined as  
\[F_{\text{max}}(v)=\begin{cases}91.7\text{ kN},&0\leqslant v\leqslant 28.5\text{ km}/\text{h},\\ 736.1v^{-1.1}\text{ kN},&28.5<v\leqslant 70.0\text{ km}/\text{h};\end{cases} \tag{1}\]  
\[B_{\text{max}}(v)=\begin{cases}91.7\text{ kN},&0\leqslant v\leqslant 37.9\text{ km}/\text{h},\\ 1114.2v^{-1.1}\text{ kN},&37.9<v\leqslant 70.0\text{ km}/\text{h}.\end{cases} \tag{2}\]  
Trams in semi-exclusive right-of-way mode enjoy independent right-of-way in running sections and share right-of-way with other road users at intersections. The speed limit can be divided into section speed limit and intersection speed limit. As shown in Figure 3, there are several feasible operating curves for trans in the running section, but only one curve represents how a tram can pass in the shortest time, which is called the fastest curve. In previous studies, to save train passengers time when traveling, the combination of maximum traction, cruise, coast and maximum braking is usually used to form the optimal control curve of the train [32].  
Factors such as the running condition of social vehicles at intersections should be considered when planning the optimal control curve of trans; thus, the traction and braking forces in the traction and braking stages need not be the maximum. The running time of the trans in the running section can be flexibly changed by adjusting the traction and braking forces; i.e., the planning time can be longer than the time of the fastest curve.  
Figure 3: Four-phase driving strategy curve of tram.  
The traction acceleration \(a_{\text{acc}}\), coast acceleration \(a_{\text{coast}}\), braking acceleration \(a_{\text{brake}}\) and basic resistance \(W\) of the trams are obtained by analyzing the traction, coast and braking states of the trams, respectively, and are expressed as follows:  
\[a_{\text{acc}}=\left(P_{1}F_{\text{max}}-W\right)/M_{\text{T}} \tag{3}\]  
\[a_{\text{coast}}=-W/M_{\text{T}} \tag{4}\]  
\[a_{\text{brake}}=-\left(P_{2}B_{\text{max}}+W\right)/M_{\text{T}} \tag{5}\]  
\[W=M_{\text{T}}g\left(A+Bv+Cv^{2}\right) \tag{6}\]  
where \(M_{\text{T}}\) represents the total mass of a fully loaded tram (set to 67.5 t); \(P_{1}\) and \(P_{2}\) are the ratio of maximum traction and maximum braking force; \(A\) is the coefficient of basic resistance (set to 2.62); \(B\) is the coefficient of rolling resistance (set to 0.094) and \(C\) is the coefficient of air resistance (set to 0.00078).  
The acceleration motion of the tram running process is a complex variable; thus, the conventional discrete method is used in the simulation process. The information storage matrix is established and the whole running process is divided into several segments of uniformly accelerated rectilinear motion with a step length \(\Delta t\) (set to 0.1 s). The discrete speed and distance of trams are calculated as follows:  
\[v_{i+1}=v_{i}+a_{i}\Delta t \tag{7}\]  
\[s_{i}=v_{i}\Delta t+\frac{1}{2}a_{i}\Delta t^{2} \tag{8}\]  
where \(v_{i}\), \(a_{i}\) and \(s_{i}\) are the initial speed, acceleration and distance at the \(i\)th time step.