## 2 Hybrid Charging Scheme
### Sequential Sector-Based Charging by Mobile Clarger
While the next location circle is concerned with the location of the mobile charger, the service sector with the radius \(R_{s}\) shown in Figure 4b is used to evaluate the amount of power received by the undercharged cluster heads, thus influencing the decision regarding the next optimal location on the next location circle. Undercharged cluster heads are considered, when making a decision on the next action, to overcharge them as quickly as possible to prepare for energy trading. The directive gain of the directional antenna can be leveraged to limit the number of undercharged cluster heads being considered in the decision of the charger movement direction. The interior angle of each service sector is 2B, and the bisector of the interior angle connects the charger and an undercharged cluster head. Therefore, sector size is increased as the antenna gain is decreased. In the case of an omnidirectional antenna, 2B is set to 360\({}^{\circ}\). Let \(\theta_{i}(t)\), \(i=1\),..., \(N_{uch}(t)\), where \(N_{uch}(t)\) is the total number of undercharged cluster heads at time \(t\), be the \(\theta\) of the \(i\)-th undercharged cluster head at time \(t\). The \(\theta_{i}(t)\) is formed by the horizontal line and the line connecting the charger and the \(i\)-th undercharged cluster head, as shown in Figure 4a. Each \(\theta_{i}(t)\) is associated with a sector. For an angle \(\theta_{i}(t)\), the number of undercharged cluster heads \(N_{ss}(\theta_{i}(t))\) within the sector, with angle range of \((\theta_{i}(t)-B)\sim(\theta_{i}(t)+B)\), is considered to determine  
Figure 4: Next location circle and service sector for charging undercharged cluster heads: (**a**) next location circle; and (**b**) sectors including undercharged cluster heads. the movement of the charger. Then, the sum of the power \(P_{ss}^{A}(\theta_{i}(t))\), received by the undercharged cluster heads within the \(i\)-th sector, is given as  
\[P_{ss}^{A}(\theta_{i}(t))=\sum_{s=1}^{N_{SS}(\theta_{i}(t))}P_{r}^{A}(\stackrel{{ \rightarrow}}{{r}}_{is}(t),\stackrel{{\rightarrow}}{{r}}_{charger}(t) +\Delta\stackrel{{\rightarrow}}{{r}}_{\theta_{i}(t)},\phi_{is}(t)) \tag{8}\]  
where \(i=1,...,N_{uch}(t)\) and \(s=1,...,N_{ss}(\theta_{i}(t))\) and \(N_{uch}(t)\geq N_{ss}(\theta_{i}(t))\). The \(\stackrel{{\rightarrow}}{{r}}_{is}(t)=(x_{is}(t)\,y_{is}(t))\) is the location of the \(s\)-th undercharged cluster head within the \(i\)-th sector and the \(\Delta\stackrel{{\rightarrow}}{{r}}_{\theta_{i}(t)}=(r_{m}\cos \theta_{i}(t)\,r_{m}\sin\theta_{i}(t))\) is the incremental vector from the location of the charger to a point on the next location circle at angle \(\theta_{i}(t)\). Note that the locations of the undercharged cluster heads are functions of time \(t\), since the undercharged cluster heads gradually become overcharged over time. The next optimal location on the next location circle in terms of optimal \(\theta_{opt}(t)\) can be mathematically expressed as  
\[ \begin{split}\theta_{opt}(t)&=\underset{\theta_{i} (t)}{\text{argmax}}P_{ss}^{A}(\theta_{i}(t))\\ &=\underset{\theta_{i}(t)}{\text{argmax}}\sum_{s=1}^{N_{SS}( \theta_{i}(t))}P_{r}^{A}(\stackrel{{\rightarrow}}{{r}}_{is}(t), \stackrel{{\rightarrow}}{{r}}_{charger}(t)+\Delta\stackrel{{ \rightarrow}}{{r}}_{\theta_{i}(t)},\phi_{is}(t))\\ &=\underset{\theta_{i}(t)}{\text{argmax}}\sum_{s=1}^{N_{SS}( \theta_{i}(t))}g(\phi_{is}(t))\frac{\alpha}{\left(d_{is}^{\theta}(t)+\beta \right)^{2}}\end{split}  \tag{9}\]  
where \(d_{is}^{\theta}(t)=\sqrt{(x_{is}(t)-x_{charger}(t)-r_{m}\cos\theta_{i}(t))^{2} +(y_{is}(t)-y_{charger}(t)-r_{m}\sin\theta_{i}(t))^{2}}\). The \(\phi_{is}(t)\) is the orientation angle of the \(s\)-th undercharged cluster head in the \(i\)-th sector, adjusted by \(\phi_{is}(t)=0^{\circ}\) when \(\phi=\theta_{i}(t)\).