# Energy Optimization in Cluster-Based Routing Protocols for Large-Area Wireless Sensor Networks
## 5 Performance Evaluation  
In the performance evaluation, we used the lengths of messages as in the LEACH code [17]: \(l_{A}=16\), \(l_{I}=16\), and \(l_{D}=8\times(500+25)\) (bits). We considered \(N=100\) or \(N=200\). We carried extensive simulations with large sizes of sensor fields, \(L=\{200\) m, \(300\) m, \(400\) m\(\}\), compared with the typical size of \(100\) m considered in most existing WSN routing protocols [3; 18; 19; 20; 21]. The unit of \(L\) and \(R\) is meters. We assumed \(S=L\), i.e., the BS is on the boundary of the field. With our derived energy function \(E(k,R)\), Nelder-Mead Simplex algorithm converged in an average of \(50\) iterations taking around \(50\) ms on a CPU running at \(3.10\) GHz. In Figure 3, we show a contour plot of \(E(k,R)\) with \(N=200\) and \(L=400\) m. We also show the locations of initial simplex \(\mathbf{v}_{1}=(5,\ 600\) m) and the final optimized simplex \(\mathbf{v}_{opt}=(6.5,\ 525\) m) after \(54\) iterations. The marks connected with dotted lines between \(\mathbf{v}_{1}\) and \(\mathbf{v}_{opt}\) show the path of simplex transformations from the iterations of N-M optimization. A closer investigation on the convergence behavior of the N-M optimization process is shown in Appendix C. We observed that the iterations have \(3\) expansions, \(9\) reflections, \(6\) outside-contractions, and \(36\) inside-contractions out of a total of \(54\) transformations. Overall, \(66\%\) of transformations are inside-contractions, which means the energy function \(E(r,R)\) has a very good shape for optimization. We evaluated our proposed optimization by simulation with the LEACH code [17] from the originator, which runs on NS2 and has been verified in the literature, to obtain higher level of objectivity of our simulation results. For each scenario, we ran \(100\) simulations with different seed numbers to obtain narrow confidence intervals. Figure 3: Contour of energy function \(E(k,R)\) (\(N=200,L=400\) m) and N-M transformation path from initial simples \(\mathbf{v}_{1}=(5,\ 600\) m) to optimized simplex \(\mathbf{v}_{opt}=(6.5,\ 525\,\mathrm{m})\). In Table 2, we compare average energy consumption per node in a round with measure (mJ/node/round), between the original LEACH [3] and the LEACH with our optimization. Each value was obtained from several hundreds of rounds with all \(N\) nodes alive in the cluster-based routing algorithms. The larger are \(N\) and \(L\), the higher is the percentage of energy savings. We also show the optimized parameters \((k_{opt},R_{opt})\) in the table as compared with the fixed values of \((k=5,R=2L)\) in the existing LEACH-based algorithms. It was observed that the optimum number \(k\) of clusters is dominated by \(N\). On the other hand, \(R\) is mainly affected by the size, \(L\), of sensor field. Figure 4 shows simulation results with \(N=200\) and \(L=400\) m. Every node had an initial energy of 10 (J). The original LEACH [3] takes \(k=5\) by rule of thumb, and \(R=2L=800\) m to make sure that the entire sensor field is covered by any sensor node. In the figure, we show the number of nodes alive over simulation time in rounds. In the original LEACH [3], the average node lifespan is 110 rounds. Our proposed optimization, shown as \"LEACH with Optimization\", takes \(k_{opt}=6.5\) and \(R_{opt}=525\) m and outperforms the original, non-optimized LEACH [3] with an average node lifespan of 178 rounds, which is an increment of 62%. A genetic algorithm based optimization, LEACH-GA [21], does not take into account the energy dissipation for control messages, and yields an optimized value of \(k=3.4\), which resulted in quick energy depletion in our simulation environment with large area sensor field. In Figure 5, we show how much our optimization improves another LEACH-based algorithm, LEACH-DT [22], which applies uneven probabilities to nodes in CH selection depending on the distance to the BS. LEACH-DT [22] uses the same parameters \(k=5\) and \(R=2L=800\) (m) as in the original LEACH.