## 3 Results and Discussion
### Simulations for Chain-Based Network
#### 3.1.4 Network Lifetime  
The transmission energy, receiving energy, and circuit energy are the main determinants which differentiate network life of WSN s. In this section, we compare the network life time of a chain-based networks that separately uses AVG, EBDKF, and proposed statistical information fusion methods. A chain-based network with 50 nodes is considered for the simulation and they are spaced in equal distances. Previously in this paper, we proved that at BER of 0.05, when considering the accuracy of deterministic system parameter, AVG and EBDKF fusion techniques will require maximum retransmissions per hop per information frame/packet to be set to 4 and 1, respectively. Therefore, here we consider the network lifetime for such a situation in order to achieve an MSE less than 0.30. In that section, we neglected the communication burden due to sending of acknowledgment frames/packets. However, here we consider the energy loss due to acknowledgment frames/packets in a retransmission allowed network. Here, \(b_{0}\) and \(a_{0}\) are set to 0.3 and 0.1, respectively. We randomly assign values between 10 mJ to 50 mJ as initial energy levels for sensor nodes using a uniform distribution. We use Equation (10) to compute the energy consumption of each link. In Equation (10), we substitute \(R_{d}=R_{a}=19.2\) kbps, \(P_{t}=200\) mJ, \(P_{c}=P_{r}=100\) mJ and \(L_{d}=L_{a}=8\). Since \(Total\ transmissions=(Total\ Retransmissions+1)\) maximum values for X and Y are set to 4, 2, and 1 for AVG, EBDKF, and proposed fusion methods, respectively. Further Voltage(V) and Current(I) values of the Central Processing Unit of wireless sensors are set to 5 V and 20 mA respectively to calculate the energy consumption due to computations using the fusion time values given in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Fusion Function** & **CPU Cycles** & **Time ((\(T_{A}\))(\(\mu\)s))** \\ \hline Average & 36 & 2.25 \\ Proposed Method & 128 & 8 \\ EBDKF & 252 & 15.75 \\ \hline \hline \end{tabular}
\end{table}
Table 2: CPU Overhead of Different Information Fusion Functions for chain-based Network.  
We run extensive simulations and record number of total active nodes at each time step for 100 data gathering cycles and replicate for 50 times to obtain Figure 11, which depicts how the average total active nodes of the network change with time for 3 different fusion techniques.  
We can see the improvement in the network life time when we employ the proposed statistical method by observing the results in Figure 11. When the proposed statistical method is employed, 59 data gathering cycles are possible prior to losing the first node in the network. But, in AVG method and EBDKF, the first node is died after 21 and four data gathering cycles, respectively. This experimental result also proves the mathematical derivation of total information transmission load comparison for three fusion techniques. There we mentioned that the information transmission of AVG and EBDKF as 3 and 9 times than proposed method for achieving network MSE less than 0.3. Further, we can observe that by 59 data gathering cycles, proposed method has lost none of its nodes, network with Average fusion had lost 28% of nodes and EBDKF has lost all of its nodes. When observing Table 2, it is very clear that highest computational efficiency belongs to Average method. However, as the proposed method does not send any retransmissions and acknowledgments, its overall energy efficiency is considerably higher than two other fusion techniques considered even though its computational complexity is moderate. The reason for EBDKF for producing least network lifetime is due to its very high computational complexity and very high number of retransmissions compared to other two fusion techniques. Therefore, the proposed method stands tall in terms of network lifetime so it is highly suitable to be used in wireless sensor networks used in smart grid environments having energy constraints.