## 3 Proposed Methodology
### Cluster Head Selection
#### 3.2.1 TCH Selection Based on Energy-Based Timer (EBT)  
According to the energy of each sensor node, a timer is allocated to the nodes. The nodes' allotted waiting times are determined by energy. High energy nodes are supported in this phase as the potential new cluster head. If not, CH is the same node with maximum transmission energy. The model description for this timer with an energy basis is as follows. If each node can determine the average energy value of its neighbours, then let us say that node I has k neighbours. \(S_{i}=\{i_{1},\,i_{2},\,i_{3},\,\ldots i_{n}\,\,\ldots i_{k_{r}}\}\), where \(i_{n}\) is the nth neighbour node. The average energy of node I is given by following Equation (8):  
\[Average\;Energy(i)=\left\{\begin{array}{ll}\frac{1}{k}\sum_{n=1}^{k}Energy( i_{n})&k>0\\ 0&k=0\end{array}\right. \tag{8}\]  
TCH is chosen using an energy-based timer from the SNs. The following equation is utilized to evaluate the energy-based waiting time value for any sensor node ID \(S_{i}\) by Equation (9).  
\[Wait\;Time\;(s_{i})=\frac{Avg\;Energy\;of\;s_{i}\;Neighbor\;node}{Energy\;of\; S_{i}} \tag{9}\]  
According to the equation above, the waiting time gets shorter as node energy rises. Less waiting time will be allotted to the node with more energy. The potential Cluster Head is chosen to be this node. Other SNs leave CH selection when they receive this message before the start of their waiting time. Selected tentative CH broadcasts a tentative CH message within its broadcast range.  
In a hierarchical fashion, Fn gathers data. Fn acts as a parent node and gathers information from its offspring. Distance (d) between parent and child nodes is equal to \(2\tau_{dt}<d<2R_{f}\) and rdt are communication and dth ranges of Fnby Equation (10).  
\[S=\left\{C\in S\;|\;2\tau_{dt}<d_{F\to C}<2R_{f}\right\} \tag{10}\]  
Figure 4: Clustering process at one coverage area.  
The total amount of data gathered at Fnis if each C transmits a packet of 1 bits by Equation (11).  
\[F_{\text{data}}=\sum_{i}^{|S|}\ell_{i} \tag{11}\]  
Following data gathering, we compute Fn's energy consumption using Equation (12) as follows:  
\[E_{F_{n}}=e_{s}+e_{t}\times R\Big{(}{\sum_{i}^{|S|}\ell_{i}+\ell}\Big{)}\phi \Big{)}+EDA+SNR \tag{12}\]  
where R is the radius, \(\varphi\) is the data aggregation factor, es is sensing energy, et is electronic energy per bit during transmission, and EDA is data aggregation energy.  
The volume of \(S1\) is evaluated as follows by Equation (13),  
\[\begin{array}{ll}V_{S1}&=\int_{0}^{r}\int_{0}^{\frac{\pi}{2}}J_{0}^{2\pi}\, \cos\phi d\theta d\phi\\ &=\pi\int_{0}(R^{2}-Z^{2})dz\\ &=\frac{2}{3}\pi R^{3}\end{array} \tag{13}\]  
Similarly, the volume of \(S2\) is evaluated as follows by Equations (14) and (15),  
\[V_{S2}=\frac{2}{3}\pi r^{3} \tag{14}\]  
\[V_{ss}=V_{S2}-V_{S1}=\frac{2}{3}\pi\Big{(}R^{3}-r^{3}\Big{)} \tag{15}\]  
The number of nodes in the bounded spherical segment is given by the following equation if the network's node density is \(\psi\) by Equations (16) and (17).  
\[N_{ss}=\frac{2}{3}\pi\psi\Big{(}R^{3}-r^{3}\Big{)} \tag{16}\]  
\[E_{F_{n}-all}^{cv}=e_{r}+\frac{2}{3}\pi\psi\Big{(}R^{3}-r^{3}\Big{)}\times\ell \tag{17}\]