## Materials and Methods
### Load Modeling for Voltage Stability Margin  
This subsection describes the measurement-based load modeling used in this study. In this model, a least-squares optimization method is applied to minimize the error between the measured and predicted values in the parameter estimation method. Because the phasor data of the load buses are obtained to establish the TE circuit, the data might be used to estimate the load modeling. It is assumed that the load model parameters estimated from the phasor data of the load buses are applicable to the VSM. Generally, the system frequency does not change significantly; thus, load fluctuations owing to small frequency variations cannot be captured accurately. Therefore, only the relationship between the load and voltage is considered in this study. The basic algorithm of the measurement-based approach is shown in Figure 10.  
In Figure 10, the measurement-based load modeling structure is determined from the static load model. For a static load model, the exponential load model and ZIP load model are used widely. In this study, the ZIP model was selected for load modeling owing to the simplicity of its estimation. The ZIP model represents the load power as a polynomial equation of the voltages. The model structure is described below. \[P=P_{0}\bigg{(}a_{2}\bigg{(}\frac{V}{V_{0}}\bigg{)}^{2}+a_{i}\bigg{(}\frac{V} {V_{0}}\bigg{)}^{1}+a_{p}\bigg{(}\frac{V}{V_{0}}\bigg{)}^{0}\bigg{)} \tag{24}\]  
where \(a_{z}+a_{i}+a_{p}=1\). \(P_{0}\) and \(V\) are the active power and voltage in the steady state, respectively. In the parameter estimation procedure, three parameters (\(a_{z},a_{i}\), and \(a_{p}\)) of the load model are calculated using the measured data as input. The objective of the parameter estimation is to match the responses of the developed model with the measured responses. The objective function for minimizing the error is as follows:  
\[\underset{p\in P}{min}E(p)=\sum_{k=1}^{N}(z_{act}(t_{k})-z_{cal}(t_{k},p))^{2} \tag{25}\]  
where \(E(p)\) is the error function of parameter \(p\); \(Z_{act}(t_{k})\) is the actual data at \(t_{k}\); \(Z_{cal}(t_{k},p)\) is the calculated data at \(t_{k}\) with parameter \(p\); \(N\) is the total number of samples; \(P\) is the parameter space. The least-squares method is applied to optimize the matching of the measured and simulated response values. The parameter space \(P\) is defined as a permissible range of all the parameters to be estimated. The minimum parameter values in Equation (25) are considered as the desired solutions. The Nose Curve Applying the Load Modeling  
The VSM in Figure 9 assumes that the loads are composed of only constant power components. However, the loads of real power systems are not a constant power but the combination of various loads. The representative model reflecting this situation is the ZIP load model. In Figure 11, each red line represents the load characteristic curve for P_0. Two operating points are characterized by the same power but different load demands z. The load characteristic curve changes by increasing  
Figure 10: Measurement-based approach load modeling algorithm. the load demand z, the system exhibits voltage instability during critical loading z_c, and the load characteristic curve becomes tangent to the PV curve. Subsequently, the voltage stability limit extends above the maximum deliverable power point. In Figure 11, to calculate the voltage stability limit, Equation (24) is transformed to  
\[P=Z\Big{(}a_{z}V^{2}+a_{i}V^{1}+a_{p}\Big{)} \tag{26}\]  
where \(Z=z*P_{0}\). To obtain the critical load demand \(z_{c}\), Equations (23) and (26) should be solved simultaneously. However, the equations are extremely complex to solve. Therefore, Equation (27) can be transformed to yield  
\[Z=\frac{P}{\Big{(}a_{z}V^{2}+a_{i}V^{1}+a_{p}\Big{)}}. \tag{27}\]  
It is assumed that \(a_{z}\), \(a_{i}\), and \(a_{p}\) are known values from the load modeling. The load modeling and \(P\) and \(Q\) are obtained from Equation (23); thereby, we can compute the set \(z\).