# Energy Optimization in Cluster-Based Routing Protocols for Large-Area Wireless Sensor Networks
## 2 Preliminaries  
### Energy Consumption Model  
We adopt the energy consumption model stated in [3]. Each sensor node is assumed to perform power control depending on the distance between the transmitter and receiver. To transmit an \(l\)-bit message over a distance \(d\) (m), a sensor node dissipates energy \(E_{T}(l,d)\) (J) by  
\[E_{T}(l,d)=\left\{ \begin{array}{ll}l(E_{e}+\epsilon_{f}d^{2}),&\text{if }d< \delta\\ l(E_{e}+\epsilon_{m}d^{4}),&\text{if }d\geq\delta,\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Meaning** \\ \hline \(N\) & total number of sensor nodes \\ \(L\) & radius of circular-shaped sensor field, (m) \\ \(S\) & distance of BS from the center of the sensor field, (m) \\ \(E_{T}(l,d)\) & energy dissipation while transmitting an \(l\)-bit message over \(d\) meters, (J) \\ \(E_{k}(l)\) & energy dissipation which receiving an \(l\)-bit message, (J) \\ \(E_{e}\) & energy consumption in the sensor circuitry per bit, (n/bit) \\ \(\epsilon_{f}\) & energy attenuation factor in free space model, (pJ/bit/m\({}^{2}\)) \\ \(\epsilon_{m}\) & energy attenuation factor in multi-path model, (pJ/bit/m\({}^{4}\)) \\ \(E_{\text{DA}}\) & energy for data aggregation per bit, (nJ/bit) \\ \(\delta\) & \(\delta=\sqrt{\epsilon_{f}/\epsilon_{m}}\approx 87.7\) (m) \\ \(T_{i,r}\) & probability that node \(i\) chooses to become a CH in round \(r\) \\ \(k\) & expected number of cluster heads in a round \\ \(R\) & transmission radius of control messages (ADV, Join-REQ, TDMA), (m) \\ \(A(r,R)\) & effective area of a node at \(r\) (m) away from the origin with transmission range \(R\), (m\({}^{2}\)) \\ \(\overline{A}(R)\) & expectation of the effective area among nodes with transmission range \(R\) \\ \(p_{k,R}^{(n)}\) & the probability that a node has \(n\) CHs in its effective area, given \(k\) and \(R\) \\ \(p_{k,R}^{(0)}\) & the probability that a node has no CHs in its effective area, given \(k\) and \(R\) \\ \(l_{A}\) & length of the advertisement message, (bits) \\ \(l_{I}\) & length of the Join request message, (bits) \\ \(l_{T}^{(k,R)}\) & length of a TDMA message, given \(k\) and \(R\), (bits) \\ \(l_{D}\) & length of the aggregated DATA message, (bits) \\ \(r_{CH}\) & the expectation of the radius of a cluster modeled as a circle, (m) \\ \(E_{T-join}\) & avg. energy by a non-CH node while sending a Join-REQ message in a round \\ \(d_{CH}\) & distance from a member node to its CH \\ \(d_{BS}\) & distance from a node to the BS \\ \(E_{T-toCH}\) & avg. energy by a member node while sending a DATA message to its CH in a round \\ \(E_{T-toBS}\) & avg. energy consumed by a non-CH, non-member node while \\ sending DATA message to the BS in a round \\ \(E_{R-ADV}\) & avg. energy by a non-CH node while receiving ADV messages in a round \\ \(E_{R-TDMA}\) & avg. energy by a non-CH node while receiving TDMA messages in a round \\ \(E_{non-CH}(k,R)\) & avg. total energy expended by all non-CH nodes in a round \\ \(E_{R-ADV}\) & avg. energy by a CH node receiving ADV in a round \\ \(E_{R-TDMA}\) & avg. energy by a CH node receiving TDMA in a round \\ \(E_{Agg}\) & avg. data-aggregation energy in a round \\ \(E_{T-CHtoBS}\) & avg. energy by a CH node transmitting aggregated data to BS in a round \\ \(E_{R-join}\) & avg. energy by a CH node receiving Join request messages in a round \\ \(E_{R-DATA}\) & avg. energy by a CH node receiving DATA messages from its member nodes in a round \\ \(E_{CH}(k,R)\) & avg. total energy expended by all CH nodes in a round \\ \(E(k,R)\) & expected amount of total energy consumption in a round \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and notations.