## 3 Results and Discussion  
### Simulations for Chain-Based Network  
1.1 Simulations for Data Accuracy of a Deterministic Constant (Slowly Varying) Parameter Having Bounded Sensor Noise and Measurements  
In this section, we compare the \(MSE_{network}\) and the \(MSE_{sink\ node}\) when the proposed statistical information fusion method; the EBDKF and AVG method are employed in the presence of measurement errors, quantization errors, and transmission errors. We run 600 iterations for a single run in order to get enough steady state measurements while there are multiple number of runs for a simulation. This algorithm is used to get \(\hat{x}_{k,i|i}^{diff}\) and then as seen in Section 3 in [44]; it is used to obtain the estimate of deterministic parameter \(\theta\), having an expectation of 1.8. One of the reasons for selecting 1.8 as the system parameter for analysis is since it is close to zero such that sensor measurements will rarely exceed sensor measurement limit. Also, it is not one of the quantized output values in the implemented quantized scheme of 8 bits as the nearest quantized output values to it are 1.79688 and 1.875. Therefore, even the highest accurate estimation of 1.80 will cause a quantization error of 0.00312 so that effect of quantization error will be reflected on the generated results. In these simulations, we generate different network conditions for different runs and in each network condition, data is gathered 600 times. The final results are obtained by averaging these simulation results. We first analyze the impact of transmission errors. A network that consists of 50 nodes is considered and the BER of a link is increased by reducing the distances between nodes. These values are varied between \(10^{-6}\) and 0.05 by changing the distance between two consecutive nodes from 24.45 m to 41.5 m. We generate heterogeneous measurement noise variances by setting \(b_{0}=0.3\) and \(a_{0}=0.1\) with variance upper bounded by 0.7. Figure 4 compares the Network MSE and MSE at sink nodes of the proposed statistical method, EBDKF, and AVG method. We can observe that the proposed statistical method outperforms AVG method for the considered BER range. The performance gap increases significantly with the BER. The proposed method also outperforms the EBDKF when the BER, due to communication errors, is greater than 0.015. For very low BER values (<10\({}^{-5}\)), the accuracy of proposed method is slightly higher than Diffusion Kalman Filtering. It can be observed from Figure 4 that for higher BER (>0.01), the EBDKF algorithm's network mean square error increases with a rising gradient. That is due to the fact that EBDKF heavily relies on multiple transmissions among neighbors for a single diffusion estimate at an iteration step which can get highly inaccurate under transmission errors. Due to that, as it is evident from Figure 4 EBDKF is superior than the proposed method in terms of information accuracy for moderate bit error rates in the range of \(10^{-5}\) to 0.0115. However, EBDKF shows large MSE under high communication errors which can occur in a Smart Grid environment. EBDKF's accuracy degrades even below average fusion method when BER is greater than 0.0175. However, on the other hand, the proposed method is aware of the communication errors and adjust the weighting coefficient automatically which is \(\sigma_{r_{k-1}}^{2}\) in Equation (24). When communication errors increase, \(\sigma_{r_{k-1}}^{2}\) also increases, so that it having a smaller weight in the fusing equation. Due to this as evident from Figure 4, MSE in proposed method increase with a very low gradient with BER. As the proposed statistical method accurately models the received information, we are able to eliminate any bias that can occur in the final result by dividing each received data with corresponding \((1-2p_{k-1})\) coefficient. Furthermore, the use of the weighting property in the conventional BLUE estimator helps to achieve the minimum variance for the received information.