## 2 Truck Platooning Speed Control Based on Multi-Objective Optimization
### Optimal Speed Computation Based on Multi-Objective Optimization  
#### 2.2.1 Objective Function  
To optimize control speed, it is necessary to set the evaluation as an objective function. In the present study, truck platooning speed is considered the control object, aiming to relieve traffic congestion at the bottleneck from the perspective of road managers who seek to improve traffic efficiency and capacity. From the perspective of truck platooning vehicles, fuel consumption must be reduced. In this study, bottleneck cell traffic flow, overall vehicle travel time, truck platooning fuel consumption, and other factors were selected as functional objectives.  
1. Bottleneck metacell traffic flow.  
To alleviate congestion within the bottleneck cell, the objective is to maximize the cell's traffic flow in the predicted time domain. The calculation formula is as follows:  
\[\sum_{k=t+1}^{k=t+4}v_{i}(k)*\rho_{i}(k)*n*L*t \tag{8}\]2. Overall vehicle travel time.  
To reduce vehicle travel time with the objective of minimizing total travel time in a predicted time domain, the calculation formula is as follows:  
\[\sum_{k=t+1}^{k=t+4}\sum_{i=1}^{i=N}(\rho_{i}(k)*n*L*t) \tag{9}\]  
where, \(N\) denotes the number of cells.
3. Truck platooning fuel consumption.  
The objective is to minimize the fuel consumption of truck platooning in the predicted time domain from a fuel consumption and cost perspective. Prior studies analyzed the fuel calculation and calibrated the relationship between VSP distribution and vehicle fuel consumption by combining the former with real vehicle operation data [27; 28]. The fuel consumption rates of specific truck models under the complete VSP interval were obtained via software analysis.  
VSP refers to the instantaneous power of a motor vehicle per unit mass in kw/t. This value--which relates to vehicle speed, acceleration, road gradient, and other factors--can be used to calculate the fuel consumption. The formula for calculating VSP is  
\[VSP=v*(a+g*grade+g*C_{R})+\frac{1}{2}*\rho_{a}*\frac{C_{D}*A}{m}*v^{3} \tag{10}\]  
where, \(v\) is the instantaneous vehicle speed in m/s; \(a\) is the instantaneous acceleration in m/s\({}^{2}\); \(g\) is the acceleration of gravity in m/s\({}^{2}\), equivalent to approximately 9.81; _grade_ is the slope of the road in %; \(C_{R}\) denotes the rolling resistance coefficient; \(\rho_{a}\) denotes ambient air density; \(C_{D}\) represents the air resistance coefficient; \(A\) denotes the transverse sectional area in m\({}^{2}\); and m represents the total mass of the vehicle in kg.  
Different vehicle types have different calculated VSP parameters. The present study considered the test truck models used in a prior study [19; 20], which are associated with the following VSP formula:  
\[VSP=v*(a+9.81*grade+0.09199)+0.000169*v^{3} \tag{11}\]  
The fuel consumption was obtained by introducing the calculated \(VSP\) value into the corresponding interval range in the table.  
\[\sum_{k=t+1}^{k=t+4}\sum_{p=1}^{p=m}VSP(v_{p}^{m}(k))*t \tag{12}\]  
where, \(m\) is the number of truck platoons. The cell where the truck platooning position \(x_{p}(k-1)\) is located at the end moment of the--\((k-1)\)th time interval is less than \(m\).