## 3 System Description
### Network Operation  
For the WSN to operate effectively, rounds are repeated over the entire network lifetime. A round is divided into two stages: the initial setup and the stable state. The clusters are created during the initial setup of each round, while data transfers to the BS occur during the stable state. The duration of the initial setup is shorter than the stable state phase. Every round starts with the initial setup. In the initial setup, clusters are created  
Figure 4: Network energy model.  
after CHs are identified. Node election likelihood, threshold perspective, node remaining energy, and the ideal number of clusters are the parameters used in CH picking.  
Because heterogeneity has a significant effect on SN networks, we use a heterogeneous network in our model. Here, three types of SNs in a heterogeneous network, named superior, medium, and normal SNs with various initial energies are considered. For superior, medium, and normal SNs, the individual initial energy equations are shown as follows:  
\[E_{sup}=E_{0}*(1+b) \tag{4}\]  
\[E_{med}=E_{0}*(1+a) \tag{5}\]  
\[E_{nrm}=E_{0} \tag{6}\]  
where \(a\) and \(b\) are the energy extra factors of medium and superior SNs, respectively. E\({}_{0}\) is normal SN energy, and E\({}_{\rm med}\) is the medium SN energy, and then E\({}_{\rm sup}\) is the superior SN energy and E\({}_{0}\) = 0.5, \(a\) = 0.45, b = 0.90.  
The node selection likelihood for the heterogeneous SNs abides by these guidelines: the superior SN's likelihood p\({}_{\rm sup}\) is larger than the medium SN's likelihood p\({}_{\rm med}\) and the medium SN's likelihood is greater than the likelihood of the normal SN's likelihood p\({}_{\rm nrm}\), implying that superior and medium SNs have a higher chance of becoming a CH.  
T(n\({}_{\rm nrm}\)), T(n\({}_{\rm med}\)), and T(n\({}_{\rm sup}\)) are the thresholds for normal, medium, and superior SNs to become CHs, respectively. The following equation can be used to compute T(n\({}_{\rm nrm}\)), T(n\({}_{\rm med}\)), and T(n\({}_{\rm sup}\)), respectively:  
\[T(n_{nrm})=\left\{\begin{array}{l l}\frac{P_{nrm}}{1-p_{nrm}(r\;mod\;\frac{1 }{p_{nrm}})}&\quad if\;\;n_{nrm}\in G\\ 0&\quad otherwise\end{array}\right. \tag{7}\]  
\[T(n_{med})=\left\{\begin{array}{l l}\frac{p_{nrm}}{1-p_{nrm}(r\;mod\;\frac{1 }{p_{nrm}})}&\quad if\;\;n_{med}\in G\\ 0&\quad otherwise\end{array}\right. \tag{8}\]  
\[T\big{(}n_{sup}\big{)}=\left\{\begin{array}{l l}\frac{p_{sup}}{1-p_{sup}(r \;mod\;\frac{1}{p_{sup}})}&\quad if\;\;n_{sup}\in G\\ 0&\quad otherwise\end{array}\right. \tag{9}\]  
where r is the iteration number currently in effect, G is the grouping of SNs that were not chosen in the previous \(\frac{1}{\text{p}_{\rm nrm}}\) iteration to be CHs.  
In (7), p\({}_{\rm nrm}\) can be substituted by p\({}_{\rm med}\) and p\({}_{\rm sup}\) to obtain T(n\({}_{\rm med}\)), T(n\({}_{\rm sup}\)). For CH selection, every SN selects a random number R\({}_{\rm n}\) from 0 and 1.  
In addition to election likelihood, the quantity of energy necessary to obtain, combine, and transfer depends on the number of SNs in the cluster [40]. As a result, the threshold prospective E\({}_{\rm th}\) is equal to  
\[E_{th}=l(E_{dec}+E_{da})\bigg{(}\frac{N_{c}}{K_{opt}}\bigg{)} \tag{10}\]  
According to [34], between 5% and 10% of the network's active SNs should be clustered for optimal energy utilization; as a result, the suitable quantity of CHs is given as  
\[K_{opt}=\sqrt{\frac{N_{c}}{2\pi}}\sqrt{\frac{\varepsilon_{fs}}{\in_{mp}}}\; \bigg{(}\frac{M}{d_{toBS}s^{2}}\bigg{)} \tag{11}\]  
where d\({}_{\rm toBS}\) is the average distance between all CH to BS, Nc is clustering active SNs.