## 2 System Model
### System Architecture  
Figure 1 illustrates a system architecture of MCSS, consisting of multiple HAPs and sensor nodes. In MCSS, the cluster composed of one HAP and sensor nodes is considered a HAP cluster (i.e., intra-cluster). The HAP is equipped with directional and omnidirectional antennas and acts as a CH in the HAP cluster. It transfers power to adjacent sensor nodes using beamforming technology that steers the beam in a specific direction.  
In addition, the HAP transmits the data collected from its cluster members (i.e., sensor nodes) to the neighboring HAP (i.e., parent HAP) so that the collected data can be sent to the root. As HAPs are always connected to a power source, they do not need to harvest energy from any other HAPs; only data are exchanged between HAPs. The sensor node is considered a battery-powered device with a single omnidirectional antenna, harvesting energy from the HAP (i.e., its CH), and transmitting data to the HAP.  
In MCSS, the amount of energy consumed by a sensor node to transmit a data packet (\(E_{Tx,timeslot}\)) is derived based on the energy consumption model in [32] as follows.  
\[E_{Tx,timeslot}=\left(P_{sleep}L_{TxXOffset}+P_{tx}L_{data}+P_{idle}L_{TsRx AckDelay}+P_{rx}L_{Ack}\right)/L_{timeslot}, \tag{1}\]  
where \(P_{sleep}\), \(P_{tx}\), \(P_{idle}\), and \(P_{rx}\) represent the amount of energy consumed when a sensor node is in sleep, transmission (\(Tx\)), idle, and reception (\(Rx\)) states during a timeslot, respectively. In addition, \(L_{TsXOffset}\) and \(L_{TsRxAckDelay}\), indicate the time intervals for the sensor node to prepare data packet transmission and for the sensor node to wait to receive an acknowledgment (\(Ack\)) from the HAP, respectively. Further, \(L_{data}\), \(L_{Ack}\), and \(L_{timeslot}\) indicate the time it takes to transmit a data packet, an \(Ack\), and the timeslot length, respectively. The HAP supplies power to the sensor nodes based on the amount of energy consumed by each sensor node derived by Equation (1), enabling the data transmission of sensor nodes and preventing their energy depletion.  
In MCSS, an IPv6 routing protocol for low-power and lossy networks (RPL) is used for routing between HAPs and sensor nodes [33]. The RPL provides path diversity by organizing a destination-oriented directed acyclic graph (DODAG) according to a routing metric called the _Rank_. The _Rank_ refers to the routing distance from a sensor node to the root and is determined by the objective function (OF). The OF of each sensor node computes the _Rank_ of each sensor node based on the expected transmission count.  
For DODAG formation, each sensor node in RPL broadcasts the DODAG information object (DIO) message containing its _Rank_ to its neighbors. Upon receiving the DIO messages, the sensor node selects its parent among the neighbor HAPs considering the _Rank_ included in the DIO messages. In MCSS, only HAPs can be selected as the parent of HAPs and sensor nodes. When the parent selection is completed, the sensor node transmits the DODAG advertisement object message toward the root. The root can recognize the new destination information and participation of the sensor node through the received DODAG advertisement object message.