## 6 Simulations and Numerical Analysis
### Simulation Results  
In the simulation, we compare the charging strategy corresponding to the minimum energy consumption routing with the one corresponding to the minimum hop routing. For minimum hops and minimum power usage routing schemes, the relay nodes' placement for Layer A is listed in Table 3. The optimal numbers and deploying locations of relay nodes in minimum energy consumption mode are calculated according to OPT-1, OPT-2 and OPT-3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node No.** & \(r_{i}\) **(kb/s)** & **Node No.** & \(r_{i}\) **(kb/s)** & **Node No.** & \(r_{i}\) **(kb/s)** \\ \hline
1 & 17 & 5 & 14 & 9 & 19 \\
2 & 16 & 6 & 18 & 10 & 19 \\
3 & 19 & 7 & 15 & 11 & 10 \\
4 & 13 & 8 & 18 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The \(r_{i}\) of each sensor node in Layer A. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Notation** & **Value** & **Notation** & **Value** & **Notation** & **Value** \\ \hline \(E_{max}\) & 10.8 kJ & \(U\) & 5 W & \(\Phi_{1}\) & 50 nJ/b \\ \(E_{min}\) & 540 J & \(\rho\) & 50 nJ/b & \(\Phi_{2}\) & 0.013 pJ/(b m\({}^{4}\)) \\ \(v\) & 5 m/s & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of parameters related to power usage. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Placement** & **No. of Relay Nodes** & **No. of Relay Nodes for Minimum Op** & **No. of Relay Nodes for Minimum Overall** & **No. of Relay Nodes for Minimum Overall** & **No. of Relay Nodes for Minimum Overall** \\  & **Routing** & **Power Usage Routing (Theoretically)** & **Power Usage Routing (Floor Number)** & **Power Usage Routing (Ceiling Number)** \\ \hline
1-2 & 6 & 8.8 & 8 & 9 \\
2-BE & 3 & 3.9 & 3 & 4 \\
3-4 & 2 & 3.2 & 3 & 4 \\
4-BE & 6 & 8.8 & 8 & 9 \\
5-6 & 3 & 3.9 & 3 & 4 \\
6-BE & 2 & 2.9 & 2 & 3 \\
7-8 & 1 & 1.5 & 1 & 2 \\
7-BE & 3 & 4.6 & 4 & 5 \\
9-10 & 9 & 13 & 13 & 14 \\
9-BE & 2 & 3.2 & 3 & 4 \\
11-BE & 3 & 4.6 & 4 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Relay nodes’ placement for both minimum hops and minimum overall energy consumption routings in Layer A. BE, branch entrance. \begin{table}
\begin{tabular}{r r r r r r r r} \hline \hline
**Node** & **Arrival** & **Recharging** & **Remaining Battery** & **Node** & **Arrival** & **Recharging** & **Remaining Battery** \\
**No. ** & **Time (s)** & **Duration (s)** & **Energy (J)** & **No. ** & **Time (s)** & **Duration (s)** & **Energy (J)** \\ \hline
56 & 437,442 & 267 & 9462 & 16 & 454,418 & 382 & 8890 \\
55 & 437,727 & 342 & 9091  
The optimal traveling path for the recharging locomotive is calculated according to Theorem 4 and is shown in Figure 6. The length of this path is 13,980 m, and the time spent on traveling around it is 2796 s.  
The orders of nodes listed in these two tables are according to their visiting orders along the traveling path, that is the first node along the roaming path of the recharging locomotive in Table 4 is the node with number 56, and the second one is the node with number 55. The second column stands for the arrival time of the recharging locomotive in the second working cycle. The third column tells us how long this node should be recharged. The forth column shows the remaining energy of each node in the second working cycle. The lengths of each working cycle for the minimum hops and minimum overall energy consumption routing are about \(4.3\times 10^{5}\) and \(7.1\times 10^{5}\) s. The \(\xi_{S}\) for both of them is 88.52% and 90.22%, which means the minimum overall energy consumption recharging scheme achieves better \(\xi_{S}\) than the minimum hop routing recharging scheme. For the nodes in the vertical tunnel, we also apply the same approach. The optimal traveling path is shown in Figure 7. The corresponding working strategies after solving OPT-9 are shown in Table 6 and Table 7. The \(\xi_{S}\) for both of them is 72.93% and 77.23%, which indicates that the minimum energy consumption recharging scheme also yields a better objective value.