## 2 Methodology
### Substation Level  
A substation consists of branches, bus sections, protection equipment (relays), measurement instruments (PMUs), and switching equipment (breakers, isolators, etc.). Each piece of equipment is required for the power system's operation and control. H-TNTP Level 2-SB shown in Figure 3 can be directly utilized to identify the substation topology. Substations are considered to be zero-impedance networks. Since Ohm's law-based current, voltage, and impedance relationships cannot be established on a zero-impedance network, a linear relationship for currents based on the substation configuration was established following the method proposed in [25]. This is shown in Figure 3 at the substation level and elaborated in Figure 4a. The proposed substation level LSE comprises two estimations: current state estimation and voltage state estimation. The current estimation is used for bad data detection in analog current measurements and topology errors; voltage-based bad data identification can be implemented using cellular computational networks (CCNs) [30]. When the bad data are detected at the substation level, the substation LSE raises a flag and entrusts the LSE to the area level. This is based on the assumption that bad data detection indicates a faulty PMU. Thus, the voltage phasor can also be a bad measurement. Thus, this avoids conducting  
Figure 4: **(a)** Flow diagram of the proposed D-LSE at the substation level considering Substation P as an example. LSE procedure flow paths A and B, measurement communication path W, and H-TNTP-PMU topology path M are referred to in Figure 3. **(b)** Flow diagram of the proposed D-LSE at the area level considering Area S as an example. LSE procedure flow paths C and D, measurement communication paths X and Y, and H-TNTP-PMU topology path N referred to in Figure 3.  
simple weighted average estimation for the voltage phasor, which does not have inherent bad data detection. The \(T_{MF}\) is the computational time for matrix formation, \(T_{MS1}\) is the computational time to solve breaker current estimation, and \(T_{MS2}\) is the computational time to solve voltage estimation using a weighted average. The area-level LSE will use the bad data detection flag and conduct the area-level LSE, thereby avoiding bad data flagged substations in the area. The current state estimation considers the circuit breaker currents as states in the LSE. Based on Kirchhoff's Current Law (KCL), the relationship between the circuit breaker current and the injection currents in the zero impedance network can be written as (4). \(z_{inj}\) refers to the current injection by each node, and \(A_{breaker}\) is the adjacent matrix that defines the relationship, which is the node connectivity matrix (NCM) of the H-TNTP Level 2. \(r_{inj}\) is the injection current measurement residue. \[[z_{inj}]=[A_{breaker}][x]+[r_{inj}] \tag{4}\]  
Based on the availability of the breaker current measurement, another linear relationship can be established for the breaker currents by relating the breaker current estimation to the measurements with an identity matrix, as shown in (5). \(z_{breaker}\) refers to the breaker current measurements, \(I\) is the identity matrix, and \(r_{breaker}\) is the breaker current measurement residue. \[[z_{breaker}]=[I][x]+[r_{breaker}] \tag{5}\]  
By combining (4) and (5), a single linear state estimation problem can be formed as (6), which again can be summarized in the form shown in (2), where the \(B\) matrix given in (7) consists of 1, 0, and \(-1\) as elements. \[[z]=\left[ \begin{array}{c}z_{inj}\\ z_{breaker}\end{array} \right]=\left[ \begin{array}{c}A_{breaker}\\ I\end{array} \right]x+\left[ \begin{array}{c}r_{inj}\\ r_{breaker}\end{array} \right] \tag{6}\]  
\[[B]=\left[ \begin{array}{c}A_{breaker}\\ I\end{array} \right] \tag{7}\]  
Voltage state estimation is conducted at the substation level if no bad data are detected. The voltage state estimation is an equal-weighted average estimation considering all available voltage measurements at the substation.