## 4 Energy Model of the Proposed Model  
A sensor is a micro-electro-mechanical system (MEMS) [44] device that consumes its limited energy in sensing the environment, performing computations, and transmitting signals. The first-order energy model is used in the proposed model similar to the existing clustering protocols [17; 18]. A considerable amount of energy is dissipated by the transmitter amplifier, indicated by \(E_{amp}\) as shown in Figure 2. Let \(d\), \(d_{o}\), and \(L\) is destination distance, reference distance and packet size respectively, and \(E_{TX}(L,d)\) is the total transmission energy consumption by a node. \(E_{fs}d^{2}\) and \(E_{amp}d^{4}\) represent free space and multipath models of wireless signals, respectively, and \(E_{elec}\) represents the energy dissipation of transmitter circuitry. The total transmission energy can be calculated as:  
\[E_{TX}(L,d)=\begin{cases}L\times E_{elec}+L\times E_{fs}d^{2},&\text{if }d<d_{o}\\ L\times E_{elec}+L\times E_{amp}d^{4},&\text{if }d\geq d_{o}\end{cases} \tag{20}\]  
After each round (\(r\)), alive nodes have probability (\(P_{i}\)) to be selected as a CH. Therefore, the expected number of CHs (\(E_{ch}\)) can be calculated as:  
\[E_{ch}=\sum_{i=1}^{N}(P_{i}(r)\times 1)=K \tag{21}\]  
where \(N\) is the total number of alive nodes and \(K\) is the number of clusters, the approximate number of nodes in each cluster is \(N_{c}=N/K\). Therefore non-CHs (\(n_{c}\)) in each cluster will be \(n_{c}=(N/K)-1\).  
If \(L_{c}\) is the data bits of one cluster, \(E_{eleRX}\) is the energy consumption in receiving signals, and \(d_{toCH}^{2}\) is the distance of nodes to their CHs, then the energy dissipated by non-CHs in a cluster can be calculated as:  
\[E_{nonCHs_{intra}}=(n_{c})(E_{eleRX}\times L_{c}+E_{afs}\times L_{c}\times d_{toCH} ^{2}) \tag{22}\]  
If \(mm_{c}\) is non-CHs that remain silent in a cluster under the described conditions, then \(k_{c}=n_{c}-mn_{c}\) will respond to CHs. The energy dissipated by \(k_{c}\) nodes is:  
\[E_{nonCHs_{intra}}=(k_{c})(E_{eleRX}\times L_{c}+E_{afs}\times L_{c}\times d_{toCH} ^{2}) \tag{23}\]  
The difference between Equation (22) and Equation (23) shows that fewer non-CH nodes may not respond to CHs, resulting in less energy dissipation in the data aggregation process. If all member nodes are responding to the CHs, then the energy dissipation in receiving and aggregating processes by a CH can be calculated as:  
\[E_{CH_{intra}}=E_{eleRX}\times L_{c}+E_{data_{A}gg} \tag{24}\]  
The energy required to transmit the aggregated data from a CH to BS can be calculated as:  
\[E_{CH_{intra}}=E_{eleTX}\times L_{c}+E_{afs}\times L_{c}\times d_{toBS}^{2} \tag{25}\]  
The total energy consumed by a cluster can be given as:  
\[E_{CT_{dissipation}}=E_{CH_{intra}}+E_{CH_{intra}}+E_{nonCHs_{intra}} \tag{26}\]  
Finally, the overall network energy consumption by \(K\) clusters can be calculated as:  
\[E_{NT_{dissipation}}=K\times E_{CT_{dissipation}} \tag{27}\]Equation (27) shows the maximum energy dissipation of a network in the first round during the network initialization. Later on, only those non-CHs will transmit their newly sensed data that justify the specified conditions, as shown in Figure 8. It is also possible that only CHs may transmit BS data when no change or minor changes occur in a cluster. In our evaluation, a variable number of transmitting nodes are observed in each cluster, depending on the user's requirements and the frequency of the network's physical changes. We reduce the energy consumption in two phases during the transmission phase in the first- and second-level nodes, first, by restricting non-cluster heads from transmitting duplicate packets to CHs as in Equation (23). Second, the CHs receive data from all or a subset of nodes in their clusters, which results in less energy consumption during computation and aggregation phases by each cluster head in the first-level and the second-level regions.