## 3 The Clustering Protocol
### Energy Consumption Model  
This paper adopts the energy consumption model proposed in [32]. In the process of information transmission, the model contains two additional models based on the transmission distance: the free space model and the multipath fading channel model.  
\(E_{T}(l,d)\) represents the energy consumed by wirelessly transmitting an \(l\)-bit message. The expression is as follows:  
\[E_{T}(l,d)=\left\{\begin{array}{ll}l\cdot E_{elc}+l\cdot\varepsilon_{fs} \cdot d^{2},&d<d_{0}\\ l\cdot E_{elc}+l\cdot\varepsilon_{mp}\cdot d^{4},&d\geq d_{0}\end{array}\right., \tag{10}\]  
In the above formula, \(E_{elc}\) represents the energy consumed per bit by the transmitter or receiver circuit, \(d\) represents the transmission distance between the transmitter and the receiver, \(\varepsilon_{fs}\) and \(\varepsilon_{mp}\) represent the energy factor per bit in the free space model and the multipath fading channel model, respectively, \(l\) represents the size of the information, and \(d_{0}\) represents the transmission distance threshold, which is calculated as follows:  
\[d_{0}=\sqrt{\frac{\varepsilon_{fs}}{\varepsilon_{mp}}} \tag{11}\]  
Figure 8: A CH distribution diagram of the protocol in this paper.  
\(E_{R}(l)\) represents the energy consumed by receiving an \(l\)-bit message. The expression is as follows:  
\[d_{0}=\sqrt{\frac{\varepsilon_{fs}}{\varepsilon_{mp}}}E_{R}(l)=l\cdot E_{elc}, \tag{12}\]  
Next we can calculate the energy consumed by the CH and the energy consumed by the CM in a cluster according to the energy consumption model. During the operation of the WSN, the CM is responsible for transmitting the information sensed from the environment to the corresponding CH. In general, the transmission distance \(d_{toCH}\) is less than \(d_{0}\). Therefore, the corresponding energy consumption \(E_{Mem}\) is calculated as follows:  
\[E_{Mem}=l\cdot E_{elc}+l\cdot\varepsilon_{fs}\cdot d_{toCH}^{2}, \tag{13}\]  
On the other hand, the CH is also responsible for receiving and fusing the information of the CMs of the cluster and finally transmitting the information after completing the fusion to the BS for decision making. The energy \(E_{CH}\) consumed by the CH is divided into three parts: the energy \(E_{CH\_R}\) consumed by receiving, the energy \(E_{CH\_FU}\) consumed by fusing, and the energy \(E_{CH\_T}\) consumed by transmitting. The formulas of the above three are shown as 14, 15, and 16.  
\[E_{CH\_R}=k\cdot l\cdot E_{elc}, \tag{14}\]  
In the above formula, \(k\) represents the number of CMs in the cluster.  
\[E_{CH\_FU}=(k+1)\cdot l\cdot E_{DA}, \tag{15}\]  
In the above formula, \(E_{DA}\) represents the energy consumption of each bit of data processed by the CH. Since the CH itself can also sense the surrounding environment information, plus the information of the \(k\cdot l\) size received from the CMs, the total information of the \((k+1)\cdot l\) size needs to be processed.  
\[E_{CH\_T}=l\cdot E_{elc}+\left\{\begin{array}{ll}l\cdot\varepsilon_{fs}\cdot d _{toBS}^{2},&d_{toBS}<d_{0}\\ l\cdot\varepsilon_{mp}\cdot d_{toBS}^{4},&d_{toBS}\geq d_{0}\end{array}\right., \tag{16}\]  
In the above formula, \(d_{toBS}\) represents the distance between the CH and the BS. After information fusion, the original information of the \((k+1)\cdot l\) size is compressed into the size of \(l\).  
Therefore, the energy \(E_{CH}\) consumed by the CH is as follows:  
\[E_{CH}=E_{CH\_R}+E_{CH\_FU}+E_{CH\_T}=k\cdot l\cdot E_{elc}+(k+1)\cdot l\cdot E _{DA}+l\cdot E_{elc}+\left\{\begin{array}{ll}l\cdot\varepsilon_{fs}\cdot d_{ ttoBS}^{2},&d_{toBS}<d_{0}\\ l\cdot\varepsilon_{mp}\cdot d_{toBS}^{4},&d_{toBS}\geq d_{0}\end{array}\right., \tag{17}\]