## 3 Simulation Results  
The size of the total service area is 100 m \(\times\) 100 m, and the number of different distributions of sensor nodes for each combination of the antenna gain and the number of sensor nodes \(N_{T}\) is 100. The clustering process and path planning are all performed by the charger, as aforementioned. All the sensor nodes are initially energy-depleted. For each antenna gain, the total charging time and the time taken for energy trading are obtained over the range of \(N_{T}\) = 100-300. While the target energy level of the cluster heads \(EL_{\text{min}}^{A}\) is adjusted according to the type of directional antenna and charging efficiency \(T_{f}\), every sensor node must be charged above the target energy level \(ET_{S}\) = 2 Joules. Other simulation parameters, such as the total service area and the parameters of power equation, Equation (2a), are set to values identical to those in [10]. Parameters \(\alpha\) and \(\beta\) of the power equation are set to 36 and 30, respectively. As the power equation is identical to that in [10], the difference in the total amount of energy provided to the sensor network is linearly proportional to (total charging time by the scheme in [10]-[total charging time by hybrid charging scheme-time taken for energy trading by hybrid charging scheme]). Note that the energy trading simply implies the re-distribution of energy provided to the sensor network. For the charging efficiency \(T_{f}\), two values are used. By considering the experimental results in [31] with an RF band 1-2 GHz, \(T_{f}\) = 0.05 is used. In addition, for comparison, \(T_{f}\) = 0.01 is used. Table 1 presents the simulation parameters. All the threshold values for the charging by the mobile charger are chosen to ensure that the sensor nodes have enough energy for successful energy trading. The \(EL_{\text{min}}^{A}\) is set higher with higher antenna gain, since a smaller number of overcharged sensor nodes is obtained while charging the cluster heads. Furthermore, the required \(EL_{\text{min}}^{A}\) with high antenna gain 12 dB is higher with a smaller number of sensor nodes because of a smaller number of overcharged sensor nodes. Other simulation parameters are as follows. The radii of the inclusion circle and service sector are set to \(R_{cl}\) = 10 m and \(R_{s}\) = 40 m, respectively. The tolerance parameter for the GSS is set to \(10^{-4}\), which corresponds to a \(3^{\circ}\) angle spacing. Figure 9 shows the total charging time \(T_{tot}\) for the hybrid charging scheme with the charging efficiency \(T_{f}\) = 0.05 compared with the results in [10]. It is seen in Figure 9 that the hybrid charging scheme is more advantageous than that in [10] in terms of charging time. Considering that the \(T_{tot}\) of the hybrid charging scheme includes the time taken for energy trading, the total amount of energy provided by the mobile charger to the sensor network is significantly smaller to achieve the same \(ET_{s}\) for all the sensor nodes. Figure 10 shows the simulation results with \(T_{f}\) = 0.01. Even with the lower charging efficiency, the hybrid charging scheme with the directional antennas still shows reduced total charging time. The variation in total charging time according to antenna gain is smaller for small  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Charging Efficiency**} & \multirow{2}{*}{\(EL_{\text{min}}^{A}\) **for 0 dB**} & \multirow{2}{*}{\(EL_{\text{min}}^{A}\) **for 6 dB**} & \multirow{2}{*}{\(EL_{\text{min}}^{A}\) **for 12 dB**} \\ \cline{5-6}  & & 3.5 J & 300 nodes & 3.5 J & 300 nodes & 4.5 J & 300 nodes \\  & 3 J & 250 nodes & 3 J & 250 nodes & 4.5 J & 250 nodes \\ \(T_{f}\) = 0.01 & 2.5 J & 200 nodes & 3 J & 200 nodes & 6 J & 200 nodes \\  & 2.5 J & 150 nodes & 3 J & 150 nodes & 7.5 J & 150 nodes \\  & 2.5 J & 100 nodes & 3 J & 100 nodes &values as compared to that in Figure 9. This is because, for the higher \(EL_{\min}^{A}\) set for \(T_{f}\) = 0.01, the time required for charging by the charger is increased.