## 2 Hybrid Charging Scheme
### Energy Trading in the Second Stage  
Energy trading begins when all of the cluster heads become overcharged. During the first charging stage, many member nodes also become overcharged. However, some member nodes are still undercharged. To bring these undercharged member nodes to the target level \(ET_{s}\), energy trading is conducted, in which all the overcharged sensor nodes, including the cluster heads, act as seller nodes  
Figure 6: Two different charging paths with different starting points: (**a**) omnidirectional antenna; and (**b**) directional antenna with 12 dB antenna gain. Figure 7: Optimal starting points with different antenna gains. Optimal starting points for directional antennas are often located on the outskirts of the total service area. and all the undercharged sensor nodes solicit energy as buyer nodes. This process achieves energy balancing for all of the sensor nodes. All the sensor nodes are assumed to employ omnidirectional antennas and are bi-directional in transacting energy. When the mobile charger finishes charging, it broadcasts a message indicating termination of charging. Then, all the undercharged member sensor nodes, knowing their SoC, send broadcast messages by using frequencies other than the charging frequency in a time slotted fashion similar to the one by [PERSON] et al. (2018), as shown in Figure 8.  
The Node ID indicates the ID number of the member node, and the EFB is the energy flag bit. If the EFB is set to 1, the node is still undercharged. The buyer nodes keep receiving energy from the seller nodes and can become overcharged. When the buyer nodes become overcharged, they broadcast the message, as in Figure 8, with the EFB set to 0. Every overcharged sensor node acts as a seller node as long as it has surplus energy. As the seller node transmits surplus energy only, it cannot become a buyer node again. During energy trading, many-to-one correspondences are established. The amount of energy increment of the \(b\)-th buyer node during the small time interval \(\Delta t_{b}\) is given by  
\[E_{b}(t+\Delta t_{b})=E_{b}(t)+\sum\limits_{i\in S}P_{r}(\overrightarrow{r}_{ b},\overrightarrow{r}_{s_{i}})\Delta t_{b} \tag{12}\]  
where \(P_{r}(\overrightarrow{r}_{b},\overrightarrow{r}_{s_{i}})\) expressed by Equation (2a) is the power received from the \(i\)-th seller node, and \(S\) indicates the set of seller nodes over the time interval \(\Delta t_{b}\) from time \(t\). Since all the seller nodes transmit energy omnidirectionally, they also receive energy from other seller nodes during energy trading. All the losses and gains of the \(i\)-th seller node during the small time interval \(\Delta t_{s}\) are given by  
\[ \begin{split}& E_{s_{i}}(t+\Delta t_{s})=E_{s_{i}}(t)-\frac{P_{0} \Delta t_{s}}{T_{f}}+\sum\limits_{j\in S^{\prime}}P_{r}(\overrightarrow{r}_{ s_{i}},\overrightarrow{r}_{s_{j}})\Delta t_{s}\\ &=E_{s_{i}}(t)-\frac{(\alpha/\beta^{2})}{T_{f}}\Delta t_{s}\cdot u (E_{s_{i}}(t)-ET_{s})+\sum\limits_{j\in S^{\prime}}\frac{\alpha}{(a_{ij}+\beta )^{2}}\Delta t_{s}\end{split}  \tag{13}\]  
where \(s_{j}\) represents the \(j\)-th seller node and the parameter \(T_{f}=(\alpha/\beta^{2})/P_{t}\), the ratio of received power with distance \(d=0\) m to transmit power \(P_{t}\) of the seller node, indicates charging efficiency. The \(P_{r0}=(\alpha/\beta^{2})\) is obtained from Equation (2a) with \(d_{j}(t)=0\) m. The function \(u(\cdot)\) is the unit step function. As the transmit power of the charger is implicit in the power Equation (2a), the charging efficiency \(T_{f}\) for RF band is adopted from ([PERSON] et al., 2018; [PERSON] et al., 2018). The \(S^{\prime}\) indicates the set of seller nodes, excluding the \(i\)-th seller node, over the time interval \(\Delta t_{s}\) from time \(t\), and the \(d_{ij}\) represents the distance between the \(i\)-th and \(j\)-th seller nodes. The new seller node loses and gains energy according to Equation (13). The sets \(S\) and \(S^{\prime}\) are updated whenever a new seller node is added.