## 3 Optimal Energy Consumption Model in HWSNs
### Coverage Cost Metric  
In order to avoid data routing through areas sparely covered by the sensor nodes, the concept of coverage cost was originally introduced into DAPR as a routing metric by [PERSON] [27]. Since then, there have been lots of studies focusing on analyzing the coverage cost metric to prolong the lifetime of a network.  
In the Coverage Preserving Protocol, [PERSON] proposed a cluster-head selection algorithm based on the coverage cost or normalized sensing coverage area of each sensor node [28]. The sensing area only covered by a node is defined as \(\phi_{0}\), and its neighbors can be defined as \(\phi(m)\):  
\[\phi(m)=\phi_{0}+\sum_{i=0}^{\infty}\frac{\phi_{i}}{i+1} \tag{14}\]  
where \(\phi_{i}\) is the sensing area of the adjacent node \(i\).  
For the simplest example, \(\phi(m)\) is calculated as:  
\[\phi(m)=\phi_{0}+\frac{\phi_{1}}{2}=(1-\frac{S(m,n)}{\pi R^{2}})+\frac{S(m,n)} {2\pi R^{2}}=1-\frac{S(m,n)}{2\pi R^{2}} \tag{15}\]where \(\phi_{1}\) is sensing area covered by node \(m\) and node \(n\), \(S(m,n)\) is the overlap rate corresponding to those two nodes. In fact, the sensing range of a specific node \(m\) is likely to overlap with several nodes, thus the calculations become complicated and one needs location information for all the nodes. They approximate all neighbor nodes as an equivalent node with an equivalent distance to the desired node \(m\). The estimation of this distance is based on the energy consumption of transmitting and receiving beacon messages to/from all the neighbor nodes.  
To calculate the coverage cost, every node \(m\) firstly calculates its coverage cost and estimates its normalized effective sensing area in the initialization and set-up phase. Since it is too complicated to calculate the exact value of \(\phi(m)\), we proposes an approximate approach based on the amount of energy consumption to transmit and receive beacon messages.  
The specific node \(m\) within a radius of \(R\) sends beacon messages to neighbors that are within the range of \(2R\). The transmission energy \(E_{Tran}\) and receiving energy \(E_{Reci}\) are calculated respectively by:  
\[\left\{\begin{array}{c}E_{Tran}=E_{Sense}+PL(2R)\\ \\ E_{Reci}=10\mathrm{log}_{10}\frac{\alpha_{i}}{10}\\ \end{array}\right. \tag{16}\]  
where \(R\) is the sensing radius of node \(m\). \(E_{Sense}\) represents the sensitivity of the radio receiver, and function \(PL\) denotes as the propagation loss for a distance. Assume that there are \(Set(m)\) neighbor nodes responding to beacon messages. Besides, \(\alpha_{i}\) is the received signal energy level of node \(i\).  
Then, the equivalent distant \(R^{\prime}\) is approximated by:  
\[R^{\prime}=2R10^{(E_{Tx}-E_{Rx})/10n} \tag{17}\]  
where \(\eta\) denotes as the path loss exponent.  
With \(\Delta=R^{\prime}/2R\), the equivalent normalized overlapping area of node \(m\) is obtained:  
\[S(m)=\frac{2[\cos^{-1}(\Delta)-\Delta\sqrt{1-\Delta^{2}}]}{\pi} \tag{18}\]  
According to Equation (14), \(\phi(m)\) can be given as \(\phi(m)=1-\frac{S(m)}{2}\), and we can obtain that the less \(\phi(m)\), the higher the probability of node \(m\) becomes the cluster head.