# An Improved Energy-Efficient Routing Protocol for Wireless Sensor Networks
## 3 Proposed Scheme
### Improved Energy-Efficient LEACH Protocol
#### 3.2.1 Energy Consumption Model of the IEE-LEACH Protocol  
The energy consumption model of the proposed IEE-LEACH protocol is introduced in this section. Unlike the LEACH protocol, in which nodes with the same initial energy are considered, each SN in  
Figure 3: The power amplifier energy consumption to total energy consumption ratio.  
the IEE-LEACH protocol is assigned a different initial energy. It is assumed that there are \(N\) nodes that are placed in an \(L\times L\) region, and the proposed IEE-LEACH protocol takes two models into accounts, i.e., the free space model and multipath model.  
When the BS is close to the nodes, the energy dissipated follows the free space model (\(d^{2}\) power loss) [17]. Thus, the energy consumption of the CH per round is  
\[E_{CH}= (\frac{N}{k}-1)*m*E_{elec}+\frac{N}{k}*m*E_{DA} \tag{6}\] \[+m*E_{elec}+m*\epsilon_{fs}*d_{toBS}^{2},\]  
where \(k\) is the number of clusters per round in the WSN, \(N/k\) is the average node of each cluster, \(E_{DA}\) is the energy consumption of the CH receiving a message of 1 bit, and \(d_{toBS}^{2}\) denotes the expected squared distance between the CH and BS [12]. Without the loss of generality, the node \(\rho(x,y)\) is distributed in an arbitrary-shaped region. It is assumed that the base station is located at location \((a,b)\). The expected squared distance between the CH and BS can be expressed as  
\[E[d_{toBS}^{2}] =\int\int((a-x)^{2}+(b-y)^{2})\rho(x,y)dxdy \tag{7}\] \[=\int\int\frac{(a-x)^{2}+(b-y)^{2}}{A}dxdy,\]  
where \(A\) is the region of distributed nodes. The energy consumption of cluster members per round is  
\[E_{nonCH}=m*E_{elec}+m*\epsilon_{fs}*d_{toCH}^{2}, \tag{8}\]  
where \(d_{toCH}^{2}\) denotes the expected squared distance between one sensor and CH, which can be expressed as  
\[E[d_{toCH}^{2}]=\int\int(x^{2}+y^{2})\rho(x,y)dxdy=\frac{L^{2}}{2\pi k}. \tag{9}\]  
The energy consumption of each cluster is given by [12]  
\[E_{cluster} =E_{CH}+(\frac{N}{k}-1)E_{nonCH}\] \[\approx E_{CH}+\frac{N}{k}E_{nonCH}. \tag{10}\]  
Thus, the energy consumption with the IEE-LEACH protocol in the WSN during each round is given by  
\[E_{round}= kE_{cluster}\] \[= m(2 NE_{elec}+NE_{DA}+k\epsilon_{fs}d_{toBS}^{2}\] \[+N\epsilon_{fs}d_{toCH}^{2}). \tag{11}\]  
The derivative of \(E_{round}\) with respect to \(k\) is equal to the optimal number of CHs, which can be expressed as  
\[k_{opt}=\frac{\sqrt{N}}{\sqrt{2\pi}}\frac{L}{d_{toBS}}. \tag{12}\]  
In addition, the energy dissipated follows the multipath model (\(d^{4}\) power loss) when the BS is far from the nodes [21]. The energy consumption of the CH node is given by \[E_{CH}= (\frac{N}{k}-1)*m*E_{dec}+\frac{N}{k}*m*E_{DA} \tag{13}\] \[+m*E_{dec}+m*\varepsilon_{mp}*d_{toBS}^{4}.\]  
The expected value of distance can be expressed as  
\[E[d_{toBS}^{4}] =\int\int\Big{(}(a-x)^{2}+(b-y)^{2}\Big{)}^{2}\rho(x,y)dxdy\] \[=\int\int\frac{\Big{(}(a-x)^{2}+(b-y)^{2}\Big{)}^{2}}{A}dxdy. \tag{14}\]  
In the multipath fading channel model, the \(E_{nonCH}\), \(d_{toCH}^{2}\) and \(E_{cluster}\) are the same as the free space channel model mentioned above. Thus, the energy consumption of the IEE-LEACH protocol in the WSN during each round can be expressed as  
\[E_{round}= kE_{cluster}\] \[= m(2 NE_{elec}+NE_{DA}+k\varepsilon_{mp}d_{toBS}^{4}\] \[+N\varepsilon_{fs}d_{toCH}^{2}). \tag{15}\]  
The derivative of \(E_{round}\) with respect to \(k\) is equal to the optimal number of CHs, which is given by [32]  
\[k_{opt}=\frac{\sqrt{N}}{\sqrt{2\pi}}\sqrt{\frac{\varepsilon_{fs}}{\varepsilon_ {mp}}}\frac{L}{d_{toBS}^{2}}. \tag{16}\]