# HCLE: Hybrid Clustering Approach for Extending WBAN Lifetime
## 3 Preliminaries
### Energy Model  
We use the first-order radio energy model as presented in Ref. [63]. This model provides mathematical equations to characterize the energy consumption during data transmission and reception processes in WBANs. The energy consumption of the transmitter and receiver is given by Equations (1) and (2), respectively:  
\[E_{Tx}(k,d)=E_{Tx-elc}\times k+E_{amp}\times\eta\times k\times d^{\eta} \tag{1}\]  
\begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Protocol** & **Parent Selection Criteria** & **Advantages** & **Limitations** & **Type of Clustering** & **Number of Sink Nodes** \\ \hline \multirow{6}{*}{HCEL} & \multirow{6}{*}{Maximum CF 2 \(CF=\frac{dist}{RE~{}5}\times RSSI  
\[E_{Rx}(k)=E_{Rx-elec}\times k. \tag{2}\]  
In Equation (1), \(E_{Tx}(k,d)\) represents the total energy consumption (in joules) required to transmit \(k\) bits between two nodes (e.g., \(v_{i}\) and \(v_{j}\)) with a separation distance of \(d\). Here, \(E_{Tx-elec}\) is the energy (in joule/bit) needed in the transmitter's circuits, while \(E_{amp}\) is the energy (in joule/bit/m\({}^{\eta}\)) consumed by the amplifier circuit in the transmitter. \(\eta\) is the path loss coefficient. Moving on to Equation (2), \(E_{Rx}(k)\) is the amount of energy consumed (in joules) by the receiver. Within this equation, \(E_{Rx-elec}\) accounts for the energy (in joule/bit) utilized in the receiver's circuits. The values for the path loss coefficient \(\eta\) in Equation (1) are provided as follows [4]:  
\[\eta=\begin{cases}2&\text{free space;}\\ 3\text{-}4&\text{line of sight (LOS);}\\ 5\text{-}7.4&\text{non-line of sight (NLOS).}\end{cases}\]  
In our work, we account for the impact of signal degradation caused by various factors in a WBAN. These factors include obstacles like the body, walls, furniture, and indoor objects that can affect the line-of-sight transmission between sensor nodes on different body parts. To incorporate this attenuation into our analysis, we utilize an attenuation factor of \(\eta=3.38\)[23]. This factor quantifies the influence of these obstacles on the strength of the transmitted signal.  
The energy parameters used in our analysis, such as \(E_{tx-elec}\), \(E_{Rx-elec}\), and \(E_{amp}\), are hardware-dependent and specific to the system under consideration. Although several transceivers are commonly used in WBANs, we specifically employ the Chipcon CC2420 transceiver in our research. Detailed energy parameters for this transceiver are provided in Table 2[4; 23; 24; 57]. By considering the impact of signal attenuation due to obstacles and utilizing the energy parameters specific to the Chipcon CC2420 transceiver, we ensure that our analysis accurately reflects the real-world performance of the WBAN system under investigation.  
To calculate the energy consumption in our proposed routing strategy, we consider the residual energy (RE) of each node in the network. This is calculated by subtracting the consumed energy from the initial energy. The energy consumed during data transmission by a sensor node is represented by Equation (1). However, for the parent node, the energy consumption includes the terms of Equation (1) and the energy consumed in the reception process, as expressed by Equation (2). This distinction arises because the parent node not only receives data from the sensor node but also forwards it to the sink.