# Efficient Position Change Algorithms for Prolonging Driving Range of a Truck Platoon
## 3 The Position Change Problem and the Algorithms
### Formulation of the Position Change Problem  
A longitudinal formation with a very close inter-vehicle spacing allows the platooning technology to reduce aerodynamic drag, reducing the fuel consumption since a quarter of the fuel consumption is caused by air drags [11]. However, each truck has an unequal distribution of fuel consumption reduction depending on its position in the platoon. For example, the first truck in the platoon has the highest fuel consumption rate among the convoy. This may lead to the situation where the platoon needs to stop to refuel the trucks more often, which increases wasted time. Hence, we see the potential in improving the effectiveness of fuel energy by making the fuel consumption rates of the trucks to be balanced as much as possible, thus achieving a longer driving distance in total since all the fuel in each truck can be consumed efficiently. Since the leading truck experiences aerodynamic drag the most and delivers the highest fuel consumption compared to the other trucks in the platoon, we ensure that trucks with more fuel should move to the front to encounter the extra air resistance while still holding the higher amount of fuel. Similarly, trucks with less fuel should move to the trailing positions where they can experience less aerodynamic drag and obtain a lower rate of fuel consumption. By changing positions periodically, the truck platooning is likely to manage a balanced amount of fuel along traveling, such that the longer total driving range can be achieved compared to the platoon without any position change. Both the two heuristic algorithms (\"FAH\" and \"TTH\") proposed in [9] try to follow this intuition. To achieve the longest driving distance, the threshold \(c\) of \"FAH\" and the time slot size \(\delta\) of \"TTH\" should be very small. In other words, the platoon needs to change the positions whenever the ordering is not the same as the ordering of the number of fuels. Thus, theoretically, the two heuristic algorithms are the same when \(c\) and \(\delta\) are very small, in which the order of the vehicles in the platoon is the order of remaining fuels at any instant. In the following, we formulate the position change problem by using the concept of time slots introduced in the \"TTH\" algorithm. We first define some notations in Table 1.  
We assume that \(g(j)\geq g(i)\) for any two positions \(i\) and \(j\) where \(j>i\). It should be noted that \(j>i\) means that the position \(i\) is in the front compared to the position \(j\). Thus, \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(V\) & Set of \(n\) vehicles in a platoon, i.e., \(\{1,2,...,n\}\) \\ \hline \multirow{2}{*}{\(t_{k}\)} & Starting time of the time slot \(k(\geq 0)\). The time is divided into multiple time slots. The length of each time slot is \(\delta(=t_{k+1}-t_{k})\). \\ \hline \(P(i,k)\) & Position of a vehicle \(i\) at time slot \(k\). \(P\) is the position change matrix. \\ \hline \(F_{i}(t)\) & Fuel amount of a truck \(i\) at time \(t\) \\ \hline \(F(t)\) & Set of remaining fuels of \(n\) trucks at time \(t\), i.e., \(\{F_{1}(t),F_{2}(t),...,F_{n}(t)\}\) \\ \hline \(r\) & Steady fuel consumption rate per unit distance \\ \hline \multirow{2}{*}{\(g(j)\)} & Fuel consumption reduction rate at a truck position \(j\), \(1\leq j\leq n\). Lower position number means a front position in the platoon \\ \hline \(s\) & Speed of the vehicles in the platoon \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. this assumption means that front positions have a smaller consumption reduction rate than rear positions. In other words, a vehicle at position \(i\) consumes more fuel than at position \(j\).