## 4 Conclusions  
Modern power system operation is becoming complex and dynamic. Linear state estimation is a promising solution for state estimation with synchrophasor measurements. However, LSE is computationally inefficient for large power systems, and the state-of-the-art transmission network topology processing limits the true potential of LSE. An efficient, resilient, and robust multilevel distributed linear state estimation based on a hierarchical transmission network topology processing that updates the network model at the PMU rate has been presented in this paper. Using an efficient and reliable H-TNTP with synchrophasor data enables the practical implementation of a multilevel D-LSE for power systems of any scale. The typical results obtained with the multilevel D-LSE implemented on the modified IEEE two-area four-machine system model and the IEEE 68 bus system model demonstrate improved computational efficiency, resiliency against bad data, and robustness against noisy measurements. In addition, LSE at the network level integrated with the H-TNTP-PMU provides redundancy to the substation-level and area-level D-LSE. Future work includes investigating and developing security assessments for bulk power systems based on H-TNTP, multilevel D-LSE, and synchrophasor measurements with optimal usage.  
Conceptualization, D.M. and G.K.V.; methodology, D.M. and G.K.V.; software, D.M. and G.K.V.; validation, D.M. and G.K.V.; formal analysis, D.M. and G.K.V.; investigation, D.M. and G.K.V.; resources, G.K.V.; data curation, D.M.; writing--original draft preparation, D.M. and G.K.V.; writing--review and editing, G.K.V.; visualization, D.M. and G.K.V.; supervision, G.K.V.; project administration, G.K.V.; funding acquisition, G.K.V. All authors have read and agreed to the published version of the manuscript.  
This study was supported in part by the National Science Foundation (NSF) of the United States, under grants CNS 2131070, ECCS 2234032 and CNS 2318612, and the Duke Energy Distinguished Professorship Endowment Fund. Any opinions, findings and conclusions, or recommendations expressed in this material are those of author(s) and do not necessarily reflect the views of NSF and Duke Energy.  
Not applicable.  
The datasets presented in this article are not readily available because the data are part of an ongoing study. Requests to access the datasets should be directed to the authors.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} BBM & Bus branch mode \\ BHA & Breaker and half arrangement \\ CCN & Cellular computational network \\ BDDBA & Double bus double breaker arrangement \\ \end{tabular}  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Factor**} & \multicolumn{3}{c}{**Level**} \\ \cline{2-4}  & **Substation** & **Area** & **Network** \\ \hline Computational Efficiency & Low & Medium & High \\ \hline Robustness & High & Medium & Medium \\ \hline Resiliency & Low & Medium & Medium \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparison of proposed D-LSE levels.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Bus ID**} & \multicolumn{2}{c}{_Measurement_} & \multicolumn{2}{c}{_True Value_} & \multicolumn{5}{c}{**Area Level**} & \multicolumn{5}{c}{**Network Level**} \\ \cline{3-13}  & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{**Voltage Estimation**} & \multirow{2}{*}{_ERF\({}_{M}\)_} & \multirow{2}{*}{_ERF\({}_{\angle}\)_} & \multicolumn{2}{c}{**Voltage Estimation**} & \multirow{2}{*}{_