# Improving Freight Traffic Efficiency at Urban Intersections Using Heavy Vehicle Platooning
## 3 Methodology
### Platoon Departure Sequence Optimization  
Optimizing traffic signal timings and reducing congestion greatly depend on the departure sequence of HCV platoons, as it plays a crucial role. The platoon-based algorithm takes into account all heavy vehicles within each platoon, enabling them to depart as a cohesive group. This approach reduces the number of vehicle departure sequences, streamlining traffic flow through the intersection.  
To determine the most efficient departure sequence, the algorithm calculates the average vehicle delay for each platoon departure sequence and selects the one with the minimum delay. Traffic signal timings are then adapted to accommodate the prioritized platoon. Equations (4)-(7) consider various factors such as distance traveled, acceleration, deceleration, and time spent waiting at intersections to find the minimum average vehicle delay.  
\[\min AD_{s}=\Sigma\big{(}D_{p,s}-V_{p}\big{)} \tag{4}\]  
The symbol \(\mathbf{AD_{s}}\) represents the average delay of all vehicles for departure sequence \(\mathbf{s}\), while \(\mathbf{D_{p,s}}\) indicates the estimated time for platoon \(\mathbf{p}\) to leave the downstream end of the intersection in sequence \(\mathbf{s}\). This estimated time is equivalent to the expected time for the final vehicle in the platoon to depart (\(\mathbf{D_{c_{s}}}\)), which can be computed using Equation (5).  
\[D_{c,s}=\left\{\begin{array}{ll}D_{c^{\prime},s}+P_{c,s}&\text{for the first vehicle in the stopped queue}\\ D_{c^{\prime},s}+\frac{1}{s_{m}}&\text{for vehicles in second position or greater}\end{array}\right. \tag{5}\]  
An HCV \(\mathbf{c}\) takes time to accelerate and pass through the intersection. This time is called the delay penalty \(\mathbf{P_{c}}\), and Equation (6) is used to calculate it. Saturation headway is referred to as \(\mathbf{S_{m}}\) for approach \(\mathbf{m}\).  
\[P_{c,s}=\frac{\sqrt{2\times a\times l}}{a} \tag{6}\]  
\[D_{c,p}=\left\{\begin{array}{ll}P_{c,s}+\frac{1}{s_{m}}\times(n-1)&\text{ If there is a queue}\\ \frac{S_{p}}{R_{p}}+\frac{1}{s_{m}}\times(n-1)&\text{ If the platoon does not stop}\end{array}\right. \tag{7}\]  
Equations (5)-(7) were employed to derive Equation (8), which plays a crucial role in the algorithm's process of optimizing traffic signal timings. Equation (8) is specifically designed to calculate the delay time of a platoon at sequence \(\mathbf{j}\). Using this equation, the algorithm can accurately estimate the delays associated with different departure sequences, enabling it to identify and prioritize the sequence with the minimum delay.  
\[D_{p,s}=n\times P_{c_{s}}+\left(\frac{n^{2}-n}{2}\times\frac{1}{S_{m}}\right) -\left(n\times l+\left(\frac{n^{2}-n}{2}\right)\times S\right)/U_{f} \tag{8}\]  
Figure 5 visually represents the departure sequence algorithm's flowchart, illustrating how the algorithm operates at each time step. The traffic signal enumerates all possible platoon departure sequences for every time step. The algorithm then evaluates each departure sequence to determine the associated average vehicle delay. Ultimately, the departure sequence with the smallest average vehicle delay is selected, ensuring the most efficient progression of platoons through the intersection. The algorithm optimizes signal timings and improves overall traffic flow by continuously adapting to the evolving traffic conditions.  
In summary, the algorithm collects data from connected HCVs to estimate and identify platoons, calculates the average vehicle delay for each departure sequence, and selects the one with the minimum delay. Subsequently, traffic signal timings are adapted to accommodate the optimized platoon departure sequence, enhancing intersection efficiency and reducing congestion.