# Efficient Position Change Algorithms for Prolonging Driving Range of a Truck Platoon
## 4 Performance Analysis and Simulation Results
Based on the simulation results so far, it is clear that if we increase \(\delta\) in TTH, we have a lower driving range and lower switching counts. Similarly, if we decrease \(\delta\), we have a higher driving range and a higher number of position changes. \"Sorted Time Slot (STS)\" helps reduce the position change counts while having the same driving range as TTH. Figure 3 shows the comparison of sequence changing count with and without STS over the number of trucks. For the case without STS shown in Figure 3a, the results are the same as that of Figure 2. Theoretically, there can be \(n!\) different position combinations. Then, the maximum number of position changes is \(n!-1\). However, not all the cases appear in the positions because the number of time slots is much less than \(n!\). Furthermore, there can be many duplicates in multiple time slots. Thus, sorting the time slots may reduce the number of position changes because we can rearrange the time slots so that duplicates join together. Figure 3b shows the switching counts with STS. It can be clearly seen that the number of position changes is much less for various time slot sizes \(\delta\). Furthermore, we achieve the optimal position change count for \(n=2\), which is just one, while still holding the longest driving distance. When \(n=2\), the position change matrix \(P\) has columns of \((1,2)\) or \((2,1)\). In STS, we sort the position change matrix by their columns, so we have only one position change from \((1,2)\) to \((2,1)\). Figure 3: Performance of STS over Number of Trucks. Figure 2: Comparison of Switching Count. The maximum number of unique position sequences for \(n\) trucks is \(n!\). For example, \(6!=720\). However, in Figure 2(b), we have less than 80 counts for \(n=6\). That is because we have only around 100 time slots when \(\delta=30\) min. If there are more time slots, then we have more combinations of position sequences, thus many position changes. To see the effect of varying \(\delta\), we also conduct the simulation under shorter \(\delta\) from 10 to 60 min each is 10 min apart. Figure 3(a) shows the number of position changes over various \(\delta\). A shorter time slot increases the number of time slots for the same amount of initial fuel. As we can expect, a smaller \(\delta\) shows a higher position change count. This shows a trade-off between the number of position changes and the total driving range (distance). To analyze the trade-off, we compute the total driving distances over \(\delta\). Figure 3(b) shows the results. Interestingly, the total distances do not differ much for different \(\delta\). The results are quite contrary to our intuition. Therefore, we further analyze the results. The case with \(\delta=10\) and \(n=2\) has 264 time slots to drive, and the remaining fuels are the same as 3.55566 L, which is not enough for the next 10 min time slot. Actually, \(264=2^{3}\times 3\times 11=6\times 44\). Thus, 264 time slots for \(\delta=10\) is 44 times slots for \(\delta=60\). It means that for \(n=2\), the two cases of \(\delta=10\) and \(\delta=60\) show the same driving distance and have same residual fuel of 3.55566 L. This is somewhat coincidental due to the specific initial amount of fuel. On average, for a time slot size \(\delta\), there can be residual amount of fuels enough to drive for \(\delta/2\) amount of time. Thus, if \(s=85\) km/h, then the average driving distance difference between \(\delta=10\) and \(\delta=60\) is about 40 km. Nonetheless, since the proposed algorithms try to reduce the differences in the fuels among the trucks, the distance difference may not be large regardless of the time slot size \(\delta\). We further investigate the performance of STS when the amounts of initial fuels are unequal. We uniformly randomly select the initial fuel amount of a truck from \([250,300]\) gallons, or \([946.35,1135.62]\) L. The value of \(\delta\) are set to be \"30 Minutes\", \"1 Hour\" and \"2 Hours\" as in previous simulations.