## 4 Proposed Algorithm
### Heterogeneous Network Energy Dissipation Model  
In the next section, we discuss a two-level energy model for heterogeneous networks. We first describe the initial total energy of each sensor node:  
\[E_{total}=\sum_{i=1}^{M}E_{0}(1+\alpha)+\sum_{i=M}^{N-M}E_{0} \tag{6}\]  
where \(M\) represents the number of nodes and is equal to \(N\) times \(m\). The symbol \(E_{0}\) represents the initial energy of the node. Specifically, a node in this study is initialized with energy \(E_{0}(1+\alpha)\), where \(\alpha\) represents a scaling factor. This factor determines the additional energy compared to the lower/initial bound \(E_{0}\). The parameter \(\alpha\) can signify either an advanced node or a super node. When considering the different levels of nodes, we expect different levels of nodes, such as advanced nodes and super nodes, to have higher battery power compared to normal nodes. Hence, the parameter \(\alpha\) represents the parameter associated with advanced and super nodes in the energy model. The energy model utilized in this study focuses on estimating the energy consumption involved in the transmission and reception of data by the sensor nodes. To illustrate the energy dissipation process during communication, the radio energy dissipation model [23; 24] is employed. Figure 1 visually represents this model, providing an overview of how energy is dissipated during the communication process. For transmitter \(E_{Tx}(l,d)\) and receiver \(E_{Rx}(l)\), when \(E_{Tx}(l,d)\) to \(E_{Rx}(l)\) sent \(l\) message, the distance of \(d\), launch amplifier \(E_{Tx}(l,d)\) needs to consume energy:  
\[E_{Tx}(n,\ d)= \begin{cases}IE_{elec}+le_{fs}d^{2},&d\leq d_{0},\\ IE_{elec}+le_{mp}d^{4},&d>d_{0},\end{cases}  \tag{7}\]  
The energy consumed by the receiver, denoted as \(E_{Rx}(l)\), when receiving an \(l\)-bit packet, depends on several factors, including the transmission distance of both ends \(d\). The threshold transmission distance, \(d_{0}\), plays a crucial role in determining the energy consumption. It is calculated as two amplifier parameters, namely \(\varepsilon_{fs}\)corresponding to the free space and \(\varepsilon_{mp}\) corresponding to the multipath model, calculated as the square root of the ratio of the two. This threshold helps determine which model should be used based on the distance between the transmitter and receiver. In energy calculation, the energy used for a single bit in electronic device sending and receiving is expressed as \(E_{elec}\). Additionally, the energy consumption is affected by the distance. Short distances are represented by \(d^{2}\), while long distances are represented by \(d^{4}\). To determine the appropriate model, the distance \(d\) is compared to the threshold \(d_{0}\). If \(d\) is less than or equal to \(d_{0}\), the free space model is employed. Conversely, if \(d\) exceeds \(d_{0}\), the multipath model is used. Taking all these factors into account, the energy used by the receiver \(E_{Rx}(l)\) for receiving the \(l\)-bit packet can be expressed through a comprehensive equation that incorporates various parameters and models associated with the transmission distance and energy consumption:  
\[E_{Rx}(n)=lE_{elec} \tag{8}\]  
Cluster heads have a vital role in receiving signals from network nodes within a cluster, and this operation requires energy consumption. Subsequently, the received signals are aggregated and transmitted to the base station (BS), which is typically located at a considerable distance from the nodes [3]. Therefore, in order to successfully transmit data to the base station, it is essential that the cluster head has sufficient energy. To represent the energy level of a cluster, various measures or formulas can be utilized. \[E_{CH}=l\cdot\frac{n}{k}\cdot(E_{elec}+E_{DA})+l\cdot\varepsilon_{mp}\cdot d_{toBS}^ {4} \tag{9}\]  
Figure 1: First-order radio energy model. The cluster head has an energy of \(E_{CH}\). In a network of sensor nodes, \(n\) is the number and \(E_{elec}\) is the energy consumed by the transmitter.