2.3.1 Longitudinal dynamics model

Consider the vehicle driving scenario as shown in Fig. 4, and analyze the forces of the vehicle$ma=F_t-F_f-F_w-F_i$(1)where m is the vehicle mass, a is the vehicle acceleration, F_t is the driving force, F_f is the rolling resistance, F_w is the air resistance, and F_i is the ramp resistance.

Figure 4 The force analysis of the vehicle.

Expanding the drive and drag forces, Eq. (1) can be expressed by the following,$ma=\frac{T_{tq}{i}_0{i}_g{\eta }_T}{r_w}-\frac{C_{D}A\rho v^2}{2}-mgf\cos \theta -mg\sin \theta$(2)where T_tq is the engine torque, i₀ is the main gear ratio, i_g is the transmission ratio, η_T is the transmission efficiency, r_w is the tire radius, C_D is the air resistance coefficient, A is the windward area, ρ is the air density, v is the vehicle speed, g is the gravitational acceleration, f is the rolling resistance coefficient, and θ is the road slope angle.

Describe the longitudinal motion of the vehicle using a third-order nonlinear model:$\{\begin{array}{l}\dot{s}(t)=v(t)\hfill \\ \frac{T_{tq}(t)i_g{i}_0{\eta }_T}{r_w}=m\dot{v}(t)+\frac{1}{2}{C}_{D}A\rho v^2(t)+\hfill \\ \tau {\dot{T}}_{tq}(t)+T_{tq}(t)=T_{des}(t)\hfill \end{array}mgf\cos \theta +mg\sin \theta$(3)where s is the longitudinal displacement of the vehicle, τ is the inertial hysteresis of the vehicle longitudinal dynamics, and ${\dot{T}}_{des}(t)$ is the desired torque.

Based on the feedback linearization theory, the linearization of Eq. (3) leads to a third-order linear model of the vehicle [2]:$\{\begin{array}{l}\dot{s}(t)=v(t)\hfill \\ \dot{v}(t)=a(t)\hfill \\ \dot{a}(t)=-{\tau }^{-1}a(t)+{\tau }^{-1}{a}_{des}(t)\hfill \end{array}$(4)

2.3.2 Lateral dynamics model

A linear two-degree-of-freedom model is used to describe the lateral dynamics of the vehicle [24], as shown in Fig. 5. In the figure, xoy is the body coordinate system, o is the center of mass of the vehicle, a and b are the lengths from the center of mass to the front and rear axes, respectively, v_x and v_y are the longitudinal and lateral vehicle speeds at the center of mass of the vehicle in the body coordinate system, δ is the front wheel rotation angle, φ is the transverse sway angle, β is the lateral deflection angle of the center of mass, α_f and α_r are the lateral deflection angles of the front and rear wheels, and F_yf and F_yr denote the lateral deflection forces of the front and rear wheels.

Figure 5 Linear two-degree-of-freedom model.

The differential equation is established by the lateral force balance and the torque balance around the z-axis:$\{\begin{array}{l}m\ddot{y}=F_{yf}\cos \delta +F_{yr}\hfill \\ I_z\ddot{\phi }=a{F}_{yf}\cos \delta -b{F}_{yr}\hfill \end{array}$(5)

where I_z represents the rotational inertia of the center of mass around the z-axis.

F_yf and F_yr can be expressed by the following equation:$\{\begin{array}{c}{F}_{yf}=-C_f{\alpha }_f\\ F_{yr}=-C_r{\alpha }_r\end{array}$(6)

where C_f and C_r denote the lateral deflection stiffness of the front and rear wheels.

According to the geometric relationship between the vehicle coordinate system and the center of mass, α_f and α_r can be expressed by the following equation:$\{\begin{array}{l}{\alpha }_f=\frac{v_y+\dot{\phi }a}{v_x}-\delta \hfill \\ {\alpha }_r=\frac{v_y-\dot{\phi }b}{v_x}\hfill \end{array}$(7)

Combining Eqs. (5–7), the differential equation of lateral motion of the vehicle can be obtained:$\{\begin{array}{l}m\left({\dot{v}}_y+v_x\dot{\phi }\right)=C_f\left(\frac{\dot{\phi }a+v_y}{v_x}-\delta \right)+C_r\left(\frac{v_y-\dot{\phi }b}{v_x}\right)\hfill \\ I_z\ddot{\phi }=a{C}_f\left(\frac{\dot{\phi }a+v_y}{v_x}-\delta \right)-b{C}_r\left(\frac{v_y-\dot{\phi }b}{v_x}\right)\hfill \end{array}$(8)

4.1.1 Lane change safety judgment

The platoon travels on the highway as shown in Fig. 13, using CPV to denote the vehicle in front of the current lane of the platoon, TPV to denote the vehicle in front of the target lane, and TFV to denote the vehicle behind the target lane, with D_CPV and D_TFV denoting the spacing between the leader vehicle (E,n) and the CPV and TPV, respectively, and D_TFV denotes the spacing between the tail vehicle (E,1) and the TFV. The platoon needs to be collision-free and at a safe distance from the three environmental vehicles during the lane change.

Figure 13 The relative position of the platoon to surrounding vehicles.

Take the leader vehicle (E,n) and CPV as an example to analyze the safety distance, as shown in Fig. 14. The CPV emergency brakes at t₀ and stops at t₁. The leader vehicle (E,n) brakes after a reaction time and stops at moment B. To ensure that no collisions occur, it is necessary to meet:$x_{CPV}(t_1)-x_{E,n}(t_2)\ge L_{CPV}$(23)where x_CPV(t₁) and x_E,n(t₂) denote the position when CPV and (E,n) stop, and L_CPV denotes the length of CPV.

Figure 14 The analysis of safety distance.

For CPV, the braking process can be considered as a constant deceleration process, A can be expressed by the following equation:$x_{CPV}(t_1)=x_{CPV}(t_0)+\frac{v_{CPV}{(t_0)}^2}{2b_{CPV}}$(24)where b_CPV denotes the maximum deceleration of CPV.

The (E,n) is ICV, the acceleration and speed of (E,n) are a_E,n(t₀) and v_E,n(t₀) at t₀. Due to the delay in data processing and control system, (E,n) can not directly enter the braking state, it needs to pass time τ_E,n to enter the emergency braking state and travel at the maximum deceleration speed. (E,n) stops at t₂ and the stop position of the (E,n) can be calculated as$\begin{array}{l}{x}_{E,n}(t_2)=x_{E,n}(t_0)+\frac{{\tau }_{E,n}}{2}\left(2v_{E,n}(t_0)+a_{E,n}(t_0)\right)+\\ \frac{{\left(v_{E,n}(t_0)+a_{E,n}(t_0){\tau }_{E,n}\right)}^2}{2b_{E,n}}\end{array}$(25)

From Eqs. (23–25), the safety distance that needs to be satisfied by (E,n) and CPV can be calculated as$\begin{array}{l}{D}_{CPV}\ge \frac{{\left(v_{E,n}(t_0)+a_{E,n}(t_0){\tau }_{E,n}\right)}^2}{2b_{E,n}}-\frac{v_{CPV}{(t_0)}^2}{2b_{CPV}}+\\ \frac{{\tau }_{E,n}}{2}\left(2v_{E,n}(t_0)+a_{E,n}(t_0)\right)\end{array}$(26)

Similarly, it can be calculated that D_TPV and D_TFV need to satisfy the following conditions:$\begin{array}{l}{D}_{TPV}\ge \frac{{\left(v_{E,n}(t_0)+a_{E,n}(t_0){\tau }_{E,n}\right)}^2}{2b_{E,n}}-\frac{v_{TPV}{(t_0)}^2}{2b_{TPV}}+\\ \frac{{\tau }_{E,n}}{2}\left(2v_{E,n}(t_0)+a_{E,n}(t_0)\right)\end{array}$(27)$\begin{array}{l}{D}_{TFV}\ge \frac{{\left(v_{TFV}(t_0)+a_{TFV}(t_0){\tau }_{TFV}\right)}^2}{2b_{TFV}}-\frac{v_{E,1}{(t_0)}^2}{2b_{E,1}}+\\ \frac{{\tau }_{TFV}}{2}\left(2v_{TFV}(t_0)+a_{TFV}(t_0)\right)\end{array}$(28)where τ_TFV denotes the reaction time of the human driver.

4.1.2 Safety downgrade strategy

When the cloud decision is to change lanes, but the vehicle side judges that it is not safe to change lanes, a safety degradation strategy is designed to ensure the safety of platoon and wait for the next cycle of cloud commands. Safe acceleration is generated using the IDM model [25]:$a_{IDM}=a_{\max }\left[1-{\left(\frac{v_e}{v_{des}}\right)}^{\delta }-{\left(\frac{s^{*}(v_e,\Delta v)}{s}\right)}^2\right]$(29)$s^{*}(v_e,\Delta v)=s_0+v_{e}T+\frac{v_e\Delta v}{2\sqrt{a_{\max }b}}$(30)where a_IDM denotes the acceleration of the target vehicle, a_max denotes the maximum acceleration of the vehicle, v_e denotes the speed of the target vehicle, v_des denotes the desired speed of the target vehicle, δ denotes the acceleration index, Δv denotes the speed difference between the target vehicle and the lead vehicle with the expression Δv=v_e–v_p, v_p denotes the speed of the preceding vehicle, s^*(v_e, Δv) denotes the desired following distance of the target vehicle, s denotes the actual following distance between the target vehicle and the preceding vehicle with the expression s=x_p–x_e–l_p, x_p and x_e denote the longitudinal position of the preceding vehicle and the target vehicle, respectively, l_p denotes the body length of the lead vehicle, s₀ denotes the stationary safety distance, T denotes the safe headway time distance, and b denotes the comfortable deceleration speed.

4.1.3 Trajectory generation

The car-following trajectory is generated based on the constant acceleration (CA) model$\{\begin{array}{l}x(t)=v_{0}t+\frac{a}{2}{t}^2\hfill \\ y(t)=0\hfill \end{array}$(31)where x(t) and y(t) denote the longitudinal and lateral displacement of the vehicle, respectively, v₀ denotes the longitudinal velocity of the vehicle at the beginning of the trajectory planning, and a denotes the desired acceleration.

The lane-changing trajectory can be described by a fifth order polynomial curve$\{\begin{array}{l}x(t)=a_5{t}^5+a_4{t}^4+a_3{t}^3+a_2{t}^2+a_{1}t+a_0\hfill \\ y(t)=b_5{t}^5+b_4{t}^4+b_3{t}^3+b_2{t}^2+b_{1}t+b_0\hfill \end{array}$(32)$\begin{array}{l}\text{s}\text{.t}\text{. }x(0)=0,y(0)=0,\dot{x}(0)=v_0,\dot{y}(0)=0，\ddot{x}(0)=a_0,\ddot{y}(0)=0\\ x(\tau )=L,y(\tau )=W,\dot{x}(\tau )=v_{\tau },\dot{y}(\tau )=0,\ddot{x}(\tau )=a_{\tau },\ddot{y}(\tau )=0\end{array}$where v₀ and a₀ denote the longitudinal velocity and acceleration of the vehicle at the beginning of the trajectory planning, which can be measured by the on-board sensors, τ denotes the lane change duration, according to statistics, the vehicle lane change on the highway is generally completed within 5 s, a shorter lane change time can improve efficiency, but will lead to poor comfort, this paper chooses a fixed lane change duration (5 s), W denotes the lane width, L, v_τ and a_τ denote the longitudinal displacement, velocity and acceleration of the vehicle at the end of the lane change, which can be calculated by integrating the acceleration sequence sent down from the cloud in discrete time:$L=v_0\tau +\Delta T^2{\displaystyle \sum _{i=1}^{N-1}{\displaystyle \sum _{j=1}^i{a}_{*}(j)+}}\frac{\Delta T^2}{2}{\displaystyle \sum _{i=1}^N{a}_{*}(i)}$(33)$v_{\tau }=v_0+{\displaystyle \sum _{i=1}^N{a}_{*}(i)}$(34)$a_{\tau }=a_{*}(N)$(35)where, a_* denotes the acceleration sequence sent down from the cloud, ΔT denotes the discrete step length of the PPLD algorithm, and N denotes the discrete time step of the lane change process, which can be expressed by the following equation:$N=\frac{\tau }{\Delta T}$(36)

The lane-changing trajectory contains 12 unknown parameters, and 13 parameters of the start state, end state and lane change duration are known to uniquely determine the trajectory.

4.2.1 Longitudinal controller

The DMPC controller is designed based on the PF communication topology. Defining the state of the ith vehicle in the platoon as x_i =[s_iv_i a_i]^T . Vehicle i receives the state information of the preceding vehicle through V2V technology, and the control input of the vehicle is obtained by solving the optimization problem based on the state of the self-vehicle as well as the state of the preceding vehicle, as shown in Fig. 15.

Figure 15 The architecture of platoon longitudinal control.

The longitudinal control objective is to follow the speed of the preceding vehicle and maintain the desired space from the preceding vehicle. Since the PPLD algorithm needs to consider the platoon length, the spacing strategy of CD is chosen in order to reduce the complexity of the decision algorithm:$d_{i,des}=d_0$(37)

Define the spacing error and speed error of the ith vehicle as follows:$\{\begin{array}{l}{e}_{i,s}=s_{i-1}-s_i-l_{i-1}-d_0\hfill \\ e_{i,v}=v_i-v_{i-1}\hfill \end{array}$(38)

According to Eq. (4) and Eq. (43), the longitudinal error differential equation is defined:$\{\begin{array}{c}{x}_i(t)=A{x}_i(t)+B{u}_i(t)+C{\omega }_i(t)\\ y_i(t)=D{x}_i(t)\end{array}$(39)where,$A=\left[\begin{array}{ccc}0& -1& 0\\ 0& 0& 1\\ 0& 0& -1/\tau \end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 1/\tau \end{array}\right],C=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],D=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$$x_i={\left[\begin{array}{ccc}{e}_{s,i}& e_{v,i}& a_i\end{array}\right]}^T{u}_i=a_{i,des}{\omega }_i=a_{i-1}{y}_i={\left[\begin{array}{cc}{e}_{i,s}& e_{i,v}\end{array}\right]}^T$where x_i denotes the state vector, u_i denotes the control input, ω_i denotes the disturbance input, and y_i denotes the system output.

The discrete state space model is created by discretizing Eq. (39):$\{\begin{array}{l}{x}_i\left(j+1|k\right)=\overline{A}{x}_i\left(j|k\right)+\overline{B}{u}_i\left(j|k\right)+\overline{C}{\omega }_i\left(j|k\right)\hfill \\ y_i\left(j|k\right)=\overline{D}{x}_i\left(j|k\right)\hfill \end{array}$(40)where, $\overline{A}=\left(I+TA\right)A,\overline{B}=TB,\overline{C}=TC,\overline{D}=D$, T denotes the discrete time step.

Define the control increment for vehicle i as $\Delta u_i(j|k)=u_i(j|k)-u_i(j-1|k)$, the system output as $y_i(j|k)={\left[\begin{array}{cc}{e}_{i,s}^j& e_{i,v}^j\end{array}\right]}^T$, and the desired output as $y_{i,des}(j|k)={\left[\begin{array}{cc}{e}_{i,s}^{j,des}& e_{i,v}^{j,des}\end{array}\right]}^T$. The goal of vehicle control is to reduce the spacing error and velocity error while avoiding large acceleration variations, which can be represented by the following cost function:$\begin{array}{l}{J}_i={{\displaystyle \sum _{j=k}^{k+N_P-1}\Vert y_i(j|k)-y_{i,des}(j|k)\Vert }}_Q^2+\\ {{\displaystyle \sum _{j=k}^{k+N_C-1}\Vert \Delta u_i(j|k)\Vert }}_R^2\\ s.t.x_i(j+1|k)=Ax(j|k)+B{u}_i(j+1|k)+C\omega (k)\\ x_i(k|k)=x_i(t)\\ y_{i,\min }\le y_i(j|k)\le y_{i,\max }\\ u_{\min }\le u_i(j|k)\le u_{\max }\\ \Delta u_{\min }\le \Delta u_i(j|k)\le \Delta u_{\max }\end{array}$(41)where Q and R denote the system error and control input weight matrix, y_i,min and y_i,max denote the minimum and maximum values of the system output, u_min and u_max denote the minimum and maximum values of the acceleration, Δu_min and Δu_max denote the minimum and maximum values of the rate of change of the acceleration.

4.2.2 Lateral controller

Based on the leader-following topology design lateral LQR controller, each vehicle in the platoon receives the path sent by the leader vehicle through V2V technology, calculates the path of the self-vehicle according to the spacing model, and combines the self-vehicle state to calculate the control input by solving the optimization problem to achieve the path tracking of the vehicle, as shown in Fig. 16.

Figure 16 The architecture of platoon lateral control.

The vehicle is projected onto the reference path and the lateral path tracking error model is established, as shown in Fig. 17. In the figure, xoy denotes the geodesic coordinate system, (x,y) denotes the vehicle center of mass coordinates, φ denotes the yaw angle, θ denotes the heading angle, ref_path denotes the reference path, (x_ref, y_ref )denotes the coordinates of the projection point, e_d denotes the distance from the vehicle center of mass to the projection point, and θ_des denotes the desired heading angle of the projection point.

Figure 17 Lateral path tracking error.

When the vehicle tracks the reference path, the spacing error is denoted using e_d and the heading error e_φ is defined as$e_{\phi }=\phi -{\theta }_{des}$(42)

The derivatives of A and B can be expressed as:$\begin{array}{l}{\dot{e}}_d=v_y+v_x{e}_{\phi }\hfill \\ {\ddot{e}}_d={\dot{v}}_y+v_x{\dot{e}}_{\phi }\hfill \\ {\dot{e}}_{\phi }=\dot{\phi }-{\dot{\theta }}_{des}\hfill \\ {\ddot{e}}_{\phi }=\ddot{\phi }\hfill \end{array}$(43)

The lateral dynamics model (8) and the lateral error (47) can be rewritten in the form of a state space with state vector $x={\left[\begin{array}{cccc}{e}_d& {\dot{e}}_d& e_{\phi }& {\dot{e}}_{\phi }\end{array}\right]}^T$, control input u=δ and disturbance ω=θ_des:$\dot{x}(t)=Ax\left(t\right)+Bu\left(t\right)+C\omega \left(t\right)$(44)式中：$\begin{array}{l}A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& \frac{C_f+C_r}{m{v}_x}& -\frac{C_f+C_r}{m}& \frac{a{C}_f-b{C}_r}{m{v}_x}\\ 0& 0& 0& 1\\ 0& \frac{a{C}_f-b{C}_r}{I_z{v}_x}& -\frac{a{C}_f-b{C}_r}{I_z}& \frac{a^2{C}_f+b^2{C}_r}{I_z{v}_x}\end{array}\right],\\ B=\left[\begin{array}{c}0\\ -\frac{C_f}{m}\\ 0\\ -\frac{a{C}_f}{I_z}\end{array}\right],C=\left[\begin{array}{c}0\\ \frac{a{C}_f-b{C}_r}{m{v}_x}+v_x\\ 0\\ \frac{a^2{C}_f+b^2{C}_r}{I_z{v}_x}\end{array}\right]\end{array}$

If the disturbance θ_des in Eq. (48) is zero, then the lateral path tracking problem can be solved analytically using LQR. For a linear system, the control objective of LQR is to choose a suitable control law u^*(t)=–Kx(t) such that the following performance indicator function takes the minimum value:$J=\frac{1}{2}{\displaystyle {\int }_0^{\infty }x{\left(t\right)}^{T}Qx\left(t\right)+u{\left(t\right)}^{T}Ru\left(t\right)dt}$(45)

The feedback control law K can be given by the following equation:$K=R^{-1}{B}^{T}P$(46)where, K=[k₁k₂k₃k₄]^T, P is the solution of Riccati equation A^TP + PA – PBR⁻¹B^TP + Q =0.

Substituting the feedback control law u^*(t) = –Kx(t) into Eq. (46) yields$\dot{x}(t)=(A-BK)x(t)+C\omega (t)$(47)

When $\dot{x}(t)=0$, the system has a steady-state error due to the presence of θ_des, x(t)≠0. To eliminate the steady-state error, it is necessary to introduce the feedforward control input δ_f, i.e:$u(t)=-Kx(t)+{\delta }_f$(48)

Substituting Eq. (48) into Eq. (46) and making $\dot{x}(t)=0$, the feedforward control input δ_f that can eliminate the steady-state error of the spacing error e_d is obtained:${\delta }_f=\frac{{\dot{\theta }}_{des}}{v_x}\left(a+b-b{k}_3-\frac{m{v}_x{}^2}{a+b}\left(\frac{b}{C_f}+\frac{a}{C_r}{k}_3-\frac{a}{C_r}\right)\right)$(49)

5.1.1 Scenario 1

As shown in Fig. 18, a platoon consisting of three vehicles travels on a two-lane highway, with the left side being the fast lane and equally spaced traffic on the road, and the right lane being the slow lane, with a slow vehicle traveling at a constant speed in front of the platoon, and the platoon will gradually be affected by the slow vehicle because the desired speed of the platoon is higher than the speed of the slow vehicle.

Figure 18 Scenario 1.

In order to verify whether the algorithm can work under different initial conditions, nine cases of examples were designed, as shown in Table 2.

5.1.2 Scenario 2

As shown in Fig. 19, there is a steady flow of traffic on a two-lane highway in both the left and right lanes. At the beginning of the simulation a platoon of three vehicles is substituted for an ambient vehicle in the right lane, and the desired speed of the platoon is higher than the desired speed of the traffic flow on the right side and lower than the desired speed of the traffic flow on the left side.

Figure 19 Scenario 2.

In order to verify the effectiveness of the algorithm under different traffic flows, five cases of examples under different traffic flows are designed, as shown in Table 3.

5.5.1 Discussion of the results of Scenario 1

Figures 20 and 21 show the lateral position and speed curves of the leader vehicle respectively, from which it can be seen that the lane change moment of CPPLC is earlier than that of Baseline, and the speed is less influenced by the slow vehicle. the lane change model of Baseline is a real-time decision, in each discrete time to determine whether the current safety criterion and the incentive criterion of the MOBIL model are satisfied at the same time, and if they are satisfied, then execute lane change. Due to the influence of the slow vehicle, Baseline has a significant speed decay before finding the time to change lanes. In contrast, CPPLC is equipped with environmental vehicle state and trend prediction model, which can find the optimal lane change timing and the corresponding acceleration sequence in advance by solving the optimization problem, making the platoon less affected by environmental vehicles and the speed decay when encountering the slow vehicle is significantly smaller than Baseline.

Figure 20 Lateral position curve of the leader vehicle: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, (f) Case 6, (g) Casec 7, (h) Case 8, and (i) Case 9.

Figure 21 Speed curve of the leader vehicle: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, (f) Case 6, (g) Casec 7, (h) Case 8, and (i) Case 9.

The first lane change of CPPLC is analyzed by Case 6. CPPLC finds the least costly subproblem by solving the optimization problem containing multiple objectives based on the prediction results and executes the corresponding acceleration sequence and lane-changing decision-making sequence. In 0–74 s, the cost of the no lane change subproblem is the lowest, and the decision of CPPLC is no lane change. As the distance between the platoon and the slow vehicle gets closer, the cost of the no-lane change subproblem gradually increases, and in 67–74 s, the acceleration sequence shows a large deceleration and the platoon speed produces a significant decay. When t=74 s, the subproblem cost corresponding to the first step lane change is the lowest, CPPLC finds the lane change timing, and the acceleration sequence becomes positive, at which time the relative position of the platoon and the surrounding vehicles is shown in Fig. 22 (a), and the target gap of the platoon lane change is the left rear gap. As time passes, the least costly subproblem continues to move forward. When t=79 s, the step 1 lane change subproblem has the lowest cost and the platoon performs a lane change, at which point the relative position of the platoon to the surrounding vehicles is shown in Fig. 22 (b). When t=84 s, the platoon has finished changing lanes, at this time the relative positions of the queue and the surrounding vehicles are shown in Fig. 22(c).

Figure 22 The relative position of the platoon and surrounding vehicles, (a) generating lane change decision (b) the beginning of platoon lane change (c) the end of platoon lane change.

From Fig. 20(f), we can find that the platoon has passed the slow vehicle sometime before the end of the simulation, and the platoon is driving freely, but the speed of CPPLC is slightly lower than Baseline. this is due to the existence of fuel consumption cost in the cost function of CPPLC, and the fuel economy of the vehicle will be improved when the vehicle speed is lower, and the two optimization objectives of expected vehicle speed and economy are in conflict, so the speed of CPPLC will be slightly lower than that of Baseline.

The tracking control effect of Case 6 is shown in Fig. 23. From figures (a)–(d), it can be seen that the leader vehicle can track the planned speed better, and the other vehicles in the platoon can track the speed of the previous vehicle better and keep a stable space from the previous vehicle. The speed error is between –0.2–0.3 m/s² and the space error is between –0.3–0.1 m. From Fig. 23 (c), it can be seen that the algorithm can achieve synchronous platoon lane change and better tracking of the path with lateral error between –0.05–0.05 m. In addition, it can be found from the figure that the tracking control of CPPLC is better than Baseline. The reason for the analysis is that the speed fluctuation of CPPLC is smaller than that of Baseline, so the error caused by the speed fluctuation will be smaller.

Figure 23 Tracking control effects: (a), (c) and (e) for Baseline; (b), (d) and (f) for CPPLC.

Table 5 gives the results of simulation Scenario 1. In terms of safety, the maximum value of TTCi for PL and PF in all cases is less than the threshold, indicating that CPPLC can ensure driving safety. In terms of fuel economy, the fuel consumption per 100 km (F) of CPPLC is lower than that of Baseline in all cases, with an average fuel saving rate of 2.52%. In terms of efficiency, Baseline performs better in Case 6 and Case 9, with CPPLC performing better in the other seven cases, with an average speed increase of 1.08% compared to Baseline.

5.5.2 Discussion of the results of Scenario 2

Figures 24 and 25 show the lateral position and velocity curves of the leader vehicle. As can be seen from the figures, the CPPLC algorithm works equally well in the traffic flow, where the platoon lane change moment is earlier than Baseline and the number and magnitude of speed decay is smaller. Since the ambient vehicles in the right lane are traveling at a speed lower than the desired speed of the platoon, Baseline generates significant speed decay before finding the lane change moment, while CPPLC can find the acceleration and lane change moment earlier by solving the optimization problem because it has the ambient vehicle state prediction model, making the platoon less affected by the ambient vehicles.

Figure 24 Lateral position curve of the leader vehicle: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, and (e) Case 5.

Figure 25 Speed curve of the leader vehicle: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, and (e) Case 5.

The maximum flow is set to 600 veh/h in simulation Scenario 2, as the control object is a platoon and synchronous lane-changing is adopted. Only when there is a large gap, the platoon has enough space and motivation for lane-changing. As seen in Fig. 25 (e), when the traffic flow reaches 600 veh/h, whether it is Baseline or CPPLC, after the platoon changes to the left lane, the platoon no longer has the maneuvering space to rejoin the right lane due to the smaller gap and slower speed of the right lane, and will keep driving in the left lane until the right lane is free of ambient traffic. It can be expected that the results will be similar to Case 5 when the traffic flow continues to increase until the platoon no longer changes lanes and loses its simulation value.

The results of simulation Scenario 2 are given in Table 6. In terms of safety, the maximum values of TTCi for PL and PF in all cases are less than the set threshold, indicating that CPPLC can ensure driving safety in all cases. In terms of fuel economy, the fuel consumption per 100 km of CPPLC is lower than that of Baseline in all cases, with an average fuel saving rate of 2.87%. In terms of efficiency, the optimization space of CPPLC is relatively small in Case 1 and Case 5 due to low or high traffic flow, so the performance of CPPLC is basically consistent with Baseline. In other cases, CPPLC is better than Baseline, and overall, CPPLC has an average speed increase of 0.81% compared to Baseline.