## 2 Hybrid Charging Scheme
### Sequential Sector-Based Charging by Mobile Clarger
In order to figure out the typical variation in \(P_{\text{sc}}(\theta)\) along \(\theta\) with an omnidirectional antenna, and the variation in \(P_{\text{ss}}^{A}(\theta)\) with a directional antenna of 12 dB antenna gain, potential variations in the next location circle are illustrated, as shown in Figure 5. The \(P_{\text{sc}}(\theta)\) along \(\theta\) in Figure 5a shows uni-modality while the \(P_{\text{ss}}^{A}(\theta)\) results in multi-modality. This is because with the omnidirectional antenna the geometrical distance of the closest undercharged cluster head basically determines the \(P_{\text{sc}}(\theta)\), whereas, with directional antennas, the orientation angles as well as the geometrical distances of the undercharged cluster heads determine the \(P_{\text{ss}}^{A}(\theta)\). Because those two variations of received power produce different characteristics, two algorithms are used to determine the optimal \(\theta\). For the omnidirectional antenna, the GSS [29] is used, which is appropriate for finding an optimal solution with a uni-modal function, whereas, for directional antennas, a sector-based search based on Equation (9) is used. Figure 6 presents the paths of the mobile charger with an omnidirectional antenna, and a directional antenna with 12 dB antenna gain, when the charger starts from two different locations with each antenna. The small red circles represent cluster heads. As shown in Figure 6a, the paths created with the omnidirectional antenna consist of smooth curves, in contrast to the line segments associated with the directional antenna. With the omnidirectional antenna, the configuration of undercharged cluster heads within the service circle can affect the direction of the next movement of the mobile charger. With uniform directive gain of omnidirectional antenna over the entire range of the orientation angle, two or more undercharged cluster heads can affect the direction of the next movement. This can cause a transition of the charger along a smooth curve. In contrast, the direction of the movement of the charger with directional antenna hardly changes, once the orientation angle \(0^{\circ}\) of the directional antenna, e.g., the orientation angle providing the highest directive gain, is set to the direction of a particular undercharged cluster head. As a result, \(T_{cm}^{A}(\mathit{PATH}_{i})\) in Equation (7a) for the directional antenna of 12 dB antenna gain is significantly smaller. Both figures show that the mobile charger  
Figure 5: Variation in the power received by undercharged cluster heads according to angle \(\theta\): (**a**) \(P_{\text{sc}}(\theta)\) with an omnidirectional antenna; and (**b**) \(P_{\text{ss}}^{A}(\theta)\) with a directional antenna of 12 dB antenna gain. moves from one cluster head to another, since this is the most efficient way to charge the cluster heads over time. The paths of the directional antenna with two starting positions are similar, while the two paths for the omnidirectional antenna show a substantial disparity. This indicates that the starting point is significantly more important for the omnidirectional antenna. In the case of the directional antenna, the charger is more likely to be guided to the closest cluster head along a straight line segment regardless of the starting point. It is worth mentioning that the optimal starting points of the mobile charger are typically located on the outskirts of the total service area, as shown in Figure 7. This is because more sensor nodes fall into the sector than in the case of a centrally located charger. However, with the omnidirectional antenna this trend becomes weak. Figure 6 shows sample starting points with directional antennas. In the simulations, all the cluster head locations were taken as starting points, and their respective total charging times were evaluated to determine the smallest one.