## 2 Hybrid Charging Scheme
### Sequential Sector-Based Charging by Mobile Clarger
The \(P_{ss}^{A}(\theta_{opt}(t))\) obtained with \(\theta_{opt}(t)\) is compared with \(P_{ss}^{A}(\theta_{opt}(t-\Delta t))\), which corresponds to the current location of the charger with optimal \(\theta_{i}(t-\Delta t)\) at \(t-\Delta t\), and the optimal action between staying and shifting is taken. The \(\max(P_{ss}^{A}(\theta_{opt}(t))\), \(P_{ss}^{A}(\theta_{opt}(t-\Delta t)))\) corresponds to the integrand of Equation (7b). The \(\phi_{is}(t)\) in Equation (9) can be obtained from \((\stackrel{{\rightarrow}}{{r}}_{i}(t)-\stackrel{{ \rightarrow}}{{r}}_{charger}(t))(\stackrel{{\rightarrow}}{{r}}_{is} (t)-\stackrel{{\rightarrow}}{{r}}_{charger}(t))^{T}\) as follows  
\[\phi_{is}(t)=\arccos\Bigg{[}\frac{\left((x_{i}(t)-x_{charger}(t))\left(y_{i}(t) -y_{charger}(t)\right)\right)\left((x_{is}(t)-x_{charger}(t))\left(y_{is}(t) -y_{charger}(t)\right)\right)^{T}}{\sqrt{\left(x_{i}(t)-x_{charger}(t)\right)^ {2}+\left(y_{i}(t)-y_{charger}(t)\right)^{2}}\sqrt{\left(x_{is}(t)-x_{charger }(t)\right)^{2}+\left(y_{is}(t)-y_{charger}(t)\right)^{2}}\Bigg{]} \tag{10}\]  
In the case of the omnidirectional antenna, Equation (9) can be modified as  
\[ \begin{split}\theta_{opt}(t)&=\underset{\theta}{ \text{argmax}}P_{\text{SC}}(t)=\underset{\theta}{\text{argmax}}\sum_{s=1}^{N_{SC }}P_{r}(\stackrel{{\rightarrow}}{{r}}_{s}(t),\stackrel{{ \rightarrow}}{{r}}_{charger}(t)+\Delta\stackrel{{\rightarrow}}{{r}}_ {\theta})\\ &=\underset{\theta}{\text{argmax}}\sum_{s=1}^{N_{SC}}\frac{\alpha }{\left(d_{s}^{\theta}(t)+\beta\right)^{2}}\end{split}  \tag{11}\]  
where \(d_{s}^{\theta}(t)=\sqrt{\left(x_{s}(t)-x_{charger}(t)-r_{m}\cos\theta\right)^{ 2}+\left(y_{s}(t)-y_{charger}(t)-r_{m}\sin\theta\right)^{2}}\). In Equation (11), the continuous angle \(\theta\) is used instead of the discrete angle \(\theta_{i}(t)\) and the dependence of the directive gain on angle \(\phi\) no longer exists. To make a decision on the next action, the service circle is considered instead of the service sector. The \(P_{sc}(\theta_{opt}(t))\) obtained with \(\theta_{opt}(t)\) is compared with \(P_{sc}(\theta_{opt}(t-\Delta t))\) and the optimal action between staying and shifting is taken. Equation (9) implies that the optimal direction of the next movement of the charger at time \(t\) is the direction toward the undercharged cluster head at the angle \(\theta_{opt}(t)\) associated with a sector, where the amount of energy to be provided to the undercharged cluster heads within the sector is the largest among all the sectors associated with all the \(\theta_{i}(t)\)s, \(i=1,\,\ldots,N_{uch}(t)\). Note here that the evaluation of the amount of energy provided to the sectors is made at the next location circle, so the next movement of the charger is to a point on the next location circle with \(\theta=\theta_{opt}(t)\). The orientation angle \(\phi_{is}(t)\) in Equation (10) represents the orientation angle of the \(s\)-th undercharged cluster head within the \(i\)-th sector. The orientation angle \(\phi_{\text{is}}(t)\) is obtained with reference to the orientation angle \(\phi_{\text{is}}(t)=0^{\circ}\) when \(\phi=\theta_{i}(t)\). With omnidirectional antenna, directive gain according to the orientation angle is uniform. Since the angle range of the sector with omnidirectional antenna is \(360^{\circ}\), as described in the paragraph before Equation (8), the sector becomes a circle and thus the \(\theta_{opt}(t)\) in Equation (11) is the direction determined by all the undercharged cluster heads, rather than fraction of undercharged cluster heads in a particular sector. As predicted by Figure 4b, the smaller beamwidth affects fewer member nodes during charging by the mobile charger, which in turn increases the time taken for energy trading in the second stage. Therefore, it is necessary to increase the target energy level \(EL^{A}_{\text{min}}\) with higher antenna gain.