## Materials and Methods  
### Power System Network Remodeling  
In this section, the method to calculate the TE impedance in a multiple bus system is explained. Modern power systems are complex; thus, to simplify, they can be classified into four groups. Generally, they are divided into the following parts: Generation, power sending, transmission, and load. Figure 1 shows the four groups conceptually.  
As shown in Figure 1, the generation part contains the several generators and transformers. The power sending part contains some loads and transformers but the most power is transferred to the load part. The transmission part contains a few loads and transformers and numerous transmission lines that interconnects the power sending and load parts. The load centers are concentrated in the load part, consuming the majority active power (\(P\)) and reactive power (\(Q\)). Every part contains multiple buses; therefore, the power systems are the multiple bus systems. To calculate the TE impedance, the power system network should be reduced to a single-bus system. A single-bus system is represented in Figure 2; it includes a single generator bus, single sending end, and single load bus.  
Figure 1: Conceptual diagram of power system divided into four groups.  
To convert the multiple-bus system to a single-bus system, the virtual buses should be defined as multiple buses. Assuming that several measurement devices are installed that can measure the voltage and current phasor such as the PMU, the multiple bus system is as presented in Figure 3.  
As shown in Figure 3, the complex power value of each bus is known because the phasors of voltage and current are already obtained from the PMUs. The current of the virtual bus is the sum of each bus current of the multiple bus system. Therefore, it can be calculated easily from the measurement data. However, the voltage of the virtual bus is difficult to derive from the voltage measurement. In a previous research, the voltage of the virtual bus was calculated by the average of each bus voltage. However, this value is intended for calculation convenience and is not mathematically accurate [47].  
Hence, a new concept with a simple formulation is introduced. As mentioned above, the complex power can be obtained using the phasors of voltage and current.  
\[\overline{S_{n}}=P_{n}+jQ_{n}=\overline{V_{n}}\overline{I_{n}} \tag{1}\]  
Assuming that the voltage, current, and complex power of the virtual bus are \(\overline{V_{\alpha\
u}}\), \(\overline{I_{\alpha\
u}}\), and \(\overline{S_{\alpha\
u}}\), respectively, the current and complex power can be expressed as follows:  
\[\overline{I_{\alpha\
u}}=\sum_{i=1}^{n}\overline{I}_{i} \tag{2}\]  
\[\overline{S_{\alpha\
u}}=\sum_{i=1}^{n}\overline{S}_{i}=\sum_{i=1}^{n} \overline{S}_{i}(P_{i}+jQ_{i}) \tag{3}\]  
where \(n\) is the number of measured bus. The complex power is obtained by multiplying the voltage phasor by the conjugate of the current phasor; therefore, Equation (4) is obtained.  
\[\overline{S_{\alpha\
u}}=\overline{V_{\alpha\
u}}\overline{I_{\alpha\
u}} \tag{4}\]  
\[\overline{V_{\alpha\
u}}=\frac{\overline{S_{\alpha\
u}}}{\overline{I_{\alpha \
u}}} \tag{5}\]  
Figure 3: Multiple bus systems with the measurement units.  
Figure 2: Configuration of virtual single bus systems.  
Using Equations (2), (3), and (5), the equation of \(\overline{V_{eq}}\) is derived as  
\[\overline{V_{eq}}=\frac{\sum_{i=1}^{n}(P_{i}+jQ_{i})}{\left\{\sum_{i=1}^{n} \overline{I_{i}}\right\}^{*}}. \tag{6}\]  
This method is possible because the measurement data are in the phasor form. If PMUs such as the SCADA/EMS system that could measure the angle of the voltage or current were not installed, this process would not be possible. The \(\overline{V_{L}}\) and \(\overline{V_{S}}\) in Figure 2 can be calculated using this method without any estimation. The \(\overline{I_{L}}\) and \(\overline{I_{S}}\) can be obtained similarly. The TE circuit is composed of combinations of these values.