## 2 Hybrid Charging Scheme
### Sequential Sector-Based Charging by Mobile Clarger  
The path taken by the mobile charger is determined by the distribution of cluster heads. The charging scheme in [10] considers all the sensor nodes in the network to find the optimal charging spots, but this approach introduces high computational complexity which grows quadratically with the number of sensor nodes. Let the objective function \(T_{cm}^{A}(PATH_{i})\) be the charging time \(T_{cm}\) taken along a path \(PATH_{i}\), parameterized by an antenna type \(A\). Then, the goal of optimization is to minimize the \(T_{cm}^{A}(PATH_{i})\) as follows  
\[T_{cm,\min}^{A}=\min_{i}T_{cm}^{A}(PATH_{i})\qquad\forall paths\] (7a) subject to \[\sum_{n=1}^{N_{PL}{\pi}{{}^{H_{i}}}}\int_{(n-1)\Delta t}^{n\Delta t}P_{r}^{A} (\overrightarrow{r}_{j},\overrightarrow{r}_{charger}(t),\,\phi_{j}^{\prime}(t ))dt\geq EL_{\min}^{A}\,j=1,...,N_{cl} \tag{7b}\]  
where the \(N_{cl}\) is the number of cluster heads and \(P_{r}^{A}(\overrightarrow{r}_{j},\overrightarrow{r}_{charger}(t),\,\phi_{j}^{ \prime}(t))\) represents the power received by the \(j\)-th cluster head located at \(\overrightarrow{r}_{j}\) transmitted from the charger at time \(t\) and \(\phi_{j}^{\prime}(t)\) is the orientation angle of the \(j\)-th cluster head with respect to the direction of charger movement. Equation (7b) shows that the travelling of the mobile charger is modeled by the concatenated discrete movements and each movement, staying at the same location or shifting to a new location, is made in every time interval \(\Delta t\). The parameter \(EL_{\min}^{A}\) represents the target energy level of the cluster heads for antenna type \(A\). As the number of seller nodes during energy trading depends on antenna type \(A\), the \(EL_{\min}^{A}\) is adjusted to ensure that the cluster heads have enough energy to transmit to the buyer nodes. The \(EL_{\min}^{A}\) set for the cluster heads is different from the target energy level \(ET_{s}\) set for all the sensor nodes. The \(EL_{\min}^{A}\) is set higher than the \(ET_{s}\) so that all the sensor nodes meet \(ET_{s}\) at the end of energy trading. Note that the charging time \(T_{cm}^{A}(PATH_{i})\) depends on the starting point of the path of the mobile charger. The \(N_{PATH_{i}}\) indicates the number of time intervals \(\Delta t\) associated with the path \(PATH_{i}\) and the \(N_{PATH_{i}}\) includes the number of time intervals during which the charger stays at the same location. When comparing Equations (3) and (7a), it is noted that the transition of the mobile  
Figure 3: Directive gain g\({}_{A}(\phi)\) of different types of directional antennas according to the orientation angle \(\phi\): (**a**) antenna gain = 6 dB; and (**b**) antenna gain = 12 dB. character is not specified by [10], whereas our approach is based on sequential movements of the charger. In addition, the charging energy is evaluated for all the sensor nodes in [10], whereas in this work it is evaluated only for the cluster heads. For each time interval \(\Delta t\), a discrete action is performed by the charger. The possible discrete actions of the mobile charger at a location are either \"staying at the same location\" or \"shifting to the next optimal location\". Figure 4a shows the \"next location circle\" on which the next optimal location of the charger is placed. The charger can stay at the same location as the next discrete movement as long as the location is optimal. The time interval \(\Delta t\) for making a decision on the next movement is set to 2.5 ms and the distance \(r_{m}\) between the current location and next optimal location is set to 0.05 m. If the next discrete movement to the next location circle occurs over a sizable time, and the location of the mobile charger is continuously shifted accordingly, the velocity of the mobile charger is (0.05 m/2.5 ms =) 72 km/h.