# Energy Optimization in Cluster-Based Routing Protocols for Large-Area Wireless Sensor Networks
## 3 Energy Consumption Analysis  
Consider the circular-shaped sensor field with radius of \(L\) (m) in Figure 1. The center of the sensor field is the origin (0, 0) of the coordinate system. Let \(S\) (\(S\geq L\)) denote distance of BS from the center of the sensor field. The base station (BS) is considered to be at the border or outside of the sensor field at coordinate (\(S\), 0). Within the sensor field, we have a node at \(r\) (\(0<r<L\)) (m) away from the origin (0, 0). Each node has transmission radius \(R\) (\(L<R<2L\)) for control messages including ADV, Join-REQ, and TDMA messages. Consider a node marked at \(r\) (\(0<r<L\)) (m) away from the origin in Figure 1. Using a dashed circle, we show the transmission radius \(R\) (m) of control messages for the marked node. Inside of the sensor field, the area covered by the transmission range of the marked node is called _effective area_ of the marked node. In the case of \(0<r<R-L\), the marked node is close to the center of the sensor field. In this case, the transmission range depicted by the dashed circle includes the entire sensor field as in Figure 2a; and the effective area of the marked node is exactly the same as the entire sensor field. However, in the case of \(R-L<r<L\), the sensor field is covered in part by the transmission range of the marked node as in Figure 2b, and the effective area of the node is the intersection of the two partially overlapping circles. By assuming symmetrical propagation channel between any two nodes, a node can hear ADV messages from other nodes which only reside in its _effective area_. The effective area \(A(r,R)\) of a node at \(r\) (m) (\(0<r<L\)) away from the origin, with transmission range \(R\), is calculated by (see Appendix A for its derivation)  
\[A(r,R)=\left\{ \begin{array}{cc}\pi L^{2},&0<r<R-L,\\ I(r,R),&R-L<r<L,\end{array} \right. \tag{4}\]  
where  
\[I(r,R)=R^{2}\cos^{-1}\left(\frac{r^{2}+R^{2}-L^{2}}{2 rR}\right)+L^{2}\cos^{-1 }\left(\frac{r^{2}+L^{2}-R^{2}}{2 rL}\right)-\frac{1}{2}\sqrt{4r^{2}L^{2}-(r^{ 2}-R^{2}+L^{2})^{2}}\]  
Figure 1: Sensor field and control-message transmission range. is the intersection area of the two partially overlapped circles. The expectation, \(\overline{A}(R)\), of the effective area is given by (see Appendix B for its derivation)  
\[\overline{A}(R) = \frac{1}{\pi L^{2}}\int_{0}^{2\pi}\int_{0}^{L}A(r,R)\,r\,\mathrm{d} r\,\mathrm{d}\theta \tag{5}\] \[= \pi(R-L)^{2}+\frac{2}{L^{2}}\int_{R-L}^{L}I(r,R)\,r\,\mathrm{d}r\] \[= 2R^{2}\cos^{-1}\left(\frac{R}{2L}\right)+L^{2}\cos^{-1}\left(1- \frac{R^{2}}{2L^{2}}\right)-\frac{(2L^{2}R+R^{3})\sqrt{4L^{2}-R^{2}}}{4L^{2}}.\]  
It is noted that Equation (5) is calculated by taking into account both the total inclusion and partial overlap cases to obtain the expectation of transmission range of an arbitrary node. Given \(k\) and \(R\), we consider the probability that a node hears ADV messages from \(n\) CHs as the probability that a node has \(n\) CHs in its effective area. We model this probability as the spatial Poisson process, which is a stationary two-dimensional Poisson point process successfully adopted in modeling wireless sensor networks [23; 24; 25]:  
\[P_{k,R}^{(n)}=\frac{\left\{\lambda\overline{A}(R)\right\}^{n}}{n!}\mathrm{e}^{ -\lambda\overline{A}(R)}, \tag{6}\]  
where the density, the number of sensor nodes per unit area,  
\[\lambda=\frac{k}{\pi L^{2}}\]  
is the average number of CHs within unit area. A node can join a cluster if it has heard from one of more CHs. Therefore, the probability that a node fails to join a cluster for the current round is given by the probability of hearing zero ADV message:  
\[P_{k,R}^{(0)}=\mathrm{e}^{-\frac{k}{\pi L^{2}}\overline{A}(R)}. \tag{7}\]  
The length of ADV, Join-REQ, TDMA, and DATA messages are denoted by \(l_{A}\), \(l_{J}\), \(l_{T}^{(k,R)}\), and \(l_{D}\), respectively (in bits). Among them, lengths \(l_{A}\), \(l_{J}\), and \(l_{D}\) are fixed.