## 4 Theoretical Analyses
### Maximizing Network Information Gathering  
When designing clustering protocols for EH-WSNs, another main objective is to maximize network information gathering, which needs to minimize the network data transmission cycle \(T\). According to Equations (18) and (26), we can realize this objective by minimizing the average energy consumption \(Eav_{1}\) of nodes in the first unit. Combining Equations (6), (7), (9)-(11), (14) and (16), \(Eav_{1}\) is calculated as follows:  
\[Eav_{1}=\frac{-2 qE_{dec}m}{\rho S}c_{1}+\frac{4 Sq\varepsilon_{fs}}{9\pi m} \frac{1}{c_{1}}+2 aqE_{dec}m+\Delta \tag{27}\]  
where \(q\) is data packet length of each node; \(m\) is the number of units; \(\rho\) is node density; \(S\) is sensor field size; \(c_{1}\) is the number of clusters in the first unit; \(a\) is data aggregation ratio; and \(\Delta\) is a constant to \(m\) and \(c_{1}\), and it is expressed as follows:  
\[\Delta=aq\varepsilon_{fs}\frac{4S}{9\pi}+q\left(\frac{1}{a}-1\right)E_{DA}-q \varepsilon_{fs}\frac{4}{9\rho\pi}+\left(2-a\right)qE_{dec} \tag{28}\]  
According to Equation (27), the derivative of \(Eav_{1}\) with respect to \(c_{1}\) can be calculated by the following equation:  
\[\frac{d}{dc_{1}}\left(Eav_{1}\right)=\frac{-2 qE_{dec}m}{\rho S}-\frac{4 Sq \varepsilon_{fs}}{9\pi m}\frac{1}{c_{1}^{2}} \tag{29}\]From Equation (29), we find \(d(Eav_{1})/dc_{1}<0\), which means \(c_{1}\) should be maximized to minimize \(Eav_{1}\). In addition, when all sensor nodes in the first unit are selected as CHs, \(c_{1}\) reaches the maximum value which can be expressed as follows:  
\[c_{1}=\rho\frac{S}{m} \tag{30}\]  
where \(S\) is sensor field size; \(\rho\) is node density and \(m\) is the number of units. According to Equations (27) and (30), we can find \(d(Eav_{1})/dm>0\), which means the number of units \(m\) should be minimized to minimize \(Eav_{1}\). As aforementioned, we adopt the free-space propagation model for the inter-cluster communication. Then the maximum inter-cluster data transmission distance \(d_{max}\) should meet the following constraint:  
\[d_{\max}\leqslant d_{0} \tag{31}\]  
where \(d_{0}\) is the distance threshold that calculated by Equation (2). We define unit size as the difference between the outer radius and inner radius of this unit. In addition, for any unit \(i\) (\(1\leqslant i\leqslant m\)), its inner radius is equal to the outer radius of unit \(i\) - 1, then its size \(Ru_{i}\) can be expressed as follows:  
\[Ru_{i}=R_{i}-R_{i-1} \tag{32}\]  
where \(R_{i}\) is the outer radius of unit \(i\) (\(R_{0}=0\)). Combining Equations (13) and (32), the size \(Ru_{i}\) of unit \(i\) (\(1\leqslant i\leqslant m\)) can be recalculated as follows:  
\[Ru_{i}=\left(\sqrt{i}-\sqrt{i-1}\right)\sqrt{\frac{S}{m\pi}} \tag{33}\]  
where \(S\) is sensor field size and \(m\) is the number of units. From Equation (33), we can find that the unit closer to BS has larger size than the one father from BS. For a CH in unit \(i\) (\(1<i\leqslant m\)), we assume it has the tendency to select its routing head from the CHs in unit \(i\)-1 toward the direction of BS. Then for the CHs in unit \(i\) (\(1\leqslant i\leqslant m\)), the one nearest the outer edge may have the longest inter-cluster data transmission distance \(d_{\max}^{(i)}\), which is estimated as follows:  
\[d_{\max}^{(i)}=Ru_{i}+Ru_{i-1} \tag{34}\]  
where \(Ru_{i-1}\) and \(Ru_{i}\) are the size of unit \(i\) - 1 and \(i\) (\(Ru_{0}=0\)). The maximum inter-cluster data transmission distance \(d_{max}\) can be expressed as follows:  
\[d_{\max}=\max\left\{d_{\max}^{(i)}\middle|i=1,2,\cdots,m\right\}=d_{\max}^{(2)} \tag{35}\]  
Combing Equations (33)-(35), we can recalculate \(d_{max}\) as follows:  
\[d_{\max}=\sqrt{\frac{2S}{m\pi}} \tag{36}\]  
where \(S\) is sensor field size and \(m\) is the number of units.