## 2 WAMS-Based Power System Model Identification  
WAMSs are measurement systems based on the transmission of analogue and/or digital information using communication systems and allowing synchronization of the measurements using a common time Reference [33]. The ability to measure voltage and current phasors in a power system created new control possibilities of monitoring the operation of a large power system from the point of view of angles and magnitudes of both voltage and frequency. The ability to do this monitoring is commonly known as wide-area monitoring (WAM). Phasor Measurement Unit (PMU) is measurement system which allows the measurement of the phasors of voltages and currents in a power system. These PMUs construct dynamic measurements which includes phasors of voltage and currents and the frequency which comes together with a satellite triggered time stamp in milli-seconds time intervals. Methods on online dynamic event detection and location which uses wide-area generator rotor frequency measurements have been presented in literature [34]. These methods combine a variance-based approach and event detection with a clustering-based approach to event location. Since WAMS provides voltage and current phasors, it is possible to express the system dynamics including the swing equation as follows  
\[\begin{split}&\frac{dx_{i}}{dt}=f_{i}\left(x_{i},I_{di},I_{qi},V_{i} \right)\\ & I_{di}=h_{di}\left(x_{i},V_{i},\theta_{i}\right)\\ & I_{qi}=h_{qi}\left(x_{i},V_{i},\theta_{i}\right)\\ &\left(P_{mi}+jQ_{mi}\right)+\left(P_{Li}+jQ_{Li}\right)=\sum V_{ i}V_{j}Y_{ik}e^{i\left(\theta_{i}-\theta_{j}-\sigma_{ik}\right)}\end{split} \tag{1}\]  
where \(P_{i},Q_{i},V_{i},I,\)_and_\(\theta_{i}\) are the active power, reactive power, bus voltage, currents, and electrical angles, respectively. \(d,q,L,m,\) and \(i\) are notations for \(d\)-axis, \(q\)-axis, loads, generation, and \(i\)th busbar, respectively.  
Where the state space representation of the above equation can be given as  
\[\begin{split}&\dot{x}\,=\,p\left(x,y,u\right)\\ & 0=q\left(x,y,u\right)\end{split} \tag{2}\]  
where \(x_{i},y_{i},\) and \(u_{i}\) are the state, output, and input vectors, respectively.  
From (2), it can be seen that \(p\) and \(q\) are some known functions. The general assumption which will be consistent in this paper is that the initial values of the system are provided by WAMS. Once the power system operator has the real-time information and measurements provided by WAMS platform, the frequency behavior can be online monitored.  
There are several model orders of synchronous machine that can be adopted for creating a suitable SFR model. It is proven that the electromechanical dynamics which are presented by swing equation are sufficient for modeling the system frequency behavior in interconnected power systems. The classical model of synchronous generator described by swing equation, integrated with suitable model of other power plant components such as governor and turbine systems can provide a suitable model of frequency dynamics. The electric network is modeled by their demand in SFR model. The SFR model realized based on WAMS is given in the next section. This model is adopted for developing a multi-stage UFLS based on WAMS technique. Then, the proposed optimization problem is solved based on the realized SFR model.