## 2 Network Model and Optimal Cluster Number Calculation
### Optimal Number of Clusters  
In general, the inter-cluster communication traffic in WSN increases as the number of clusters increases, and the intra-cluster communication traffic increases as the number of clusters decreases. In addition, the network's energy consumption increases as communication traffic increases. The determination of the optimal cluster number of the network is of great significance to the network's communication. In this section, we will determine the optimal number of clusters \(k\) in combination with the network structure model and energy consumption model described respectively in Sections 2.1 and 2.2. The monitoring area in this paper is a circle with a diameter of \(M\). In real life, the cluster areas of WSN must be irregular and inconsistent, and the nodes are randomly placed. If these three points are both considered, the proposed model must be complex and not universal. So like the optimal numbers of clusters in [11; 12], that in this paper is also used as a relatively common reference standard in the actual monitoring area. To derive the optimal number of clusters more intuitively, we construct an inline square in the circular region with a side length of \(L\). We assume that the clusters in the square are all circular in shape with a radius of \(R\) and the cluster distribution is uniformly distributed. Finally, after calculating the total circular cluster area and the circular area to establish the relationship, we can obtain the relationship between the total number of clusters \(k\) and the number of circular clusters \(k_{1}\). As shown in Figure 2, the monitoring area in this paper is a large circle with a diameter of \(M\) and the length of the embedded square is \(L\). From this, we can derive the relationship between \(L\) and \(M\):  
\[L=\sqrt{2}M/_{2} \tag{6}\]  
After the sensor network is divided into many clusters, the CH receives the information transmitted by the cluster member nodes, and after processing, eventually transmits it to the BS for final data fusion. In the intra-cluster communication, as the distance between the cluster member node and the CH is not large, we adopt the free space model. To understand the network structure model more intuitively, Figure 3 shows an example dividing the cluster into 16 clusters. In the figure, the positive center position of the monitoring area is the BS indicated by \(\
ircledtriangle\). A blue circle indicates a cluster. Consequently, we can obtain the expression of the blue cluster number \(k_{1}\) as follows:  
\[k_{1}=\frac{M^{2}}{8R^{2}} \tag{7}\]  
Figure 2: Monitoring area. The area of the monitoring area \(S_{sum}\) is calculated as follows:  
\[S_{sum}=\pi\cdot\left(\frac{M}{2}\right)^{2}=\frac{\pi M^{2}}{4} \tag{8}\]  
The inscribed square area \(S_{square}\) is calculated as follows:  
\[S_{square}=\left(\frac{\sqrt{2}}{2}M\right)^{2}=\frac{M^{2}}{2} \tag{9}\]  
One blue cluster area \(S_{cluster}\) is calculated as follows:  
\[S_{cluster}=\pi R^{2}=\pi\left(\
icefrac{{\sqrt{2}}}{{2}}M/2\sqrt{k_{1}} \right)^{2}=\frac{\pi M^{2}}{8k_{1}} \tag{10}\]  
The total area of the blue clusters \(S_{cluster\_sum}\) is calculated as follows:  
\[S_{cluster\_sum}=k_{1}S_{cluster}=\frac{\pi M^{2}}{8} \tag{11}\]  
According to Equations (8) and (11):  
\[S_{sum}=2S_{cluster\_sum} \tag{12}\]  
If the cluster is completely divided into clusters in the monitoring area, the total number of clusters \(k\) is twice that of \(k_{1}\), namely:  
\[k=2k_{1}=\frac{M^{2}}{4R^{2}} \tag{13}\]  
The cluster member nodes obey the uniform distribution, and then the distribution function can be expressed as follows:  
\[g(k)=\frac{8k_{1}}{\pi M^{2}}=\frac{4k}{\pi M^{2}} \tag{14}\]  
Calculate the expected squared distance of the cluster member nodes to the CH. \[ \begin{array}{l}E[d_{toCH}^{2}]=\int\limits_{0}^{2\pi}d\theta\int\limits_{0 }^{R}g(k)\rho^{3}d\rho\\ =\int\limits_{0}^{2\pi}d\theta\int\limits_{R}^{R}\frac{4k}{\pi M^{2}}\rho^{3}d \rho\\ =\frac{2 kR^{4}}{M^{2}}\end{array}