## 3 Frequency Model Set Up  
In the control center of modern power systems, a suitable frequency model can be created based on the data provided from WAMS. From (2), the frequency response model which can be used to design an optimal UFLS scheme, can be set up based on the swing equation considering the classical model of each generator [3]. The basic idea on analyzing the power system dynamics is based on theswing equation of each synchronous generating unit which is given in (3). This equation relates the change in frequency over time to mechanical and electrical power difference with the inertia of \(i\)th power plant. \[\frac{dw_{i}}{dt}=\frac{1}{2H_{i}}\left(P_{m_{i}}-P_{e_{i}}\right) \tag{3}\]  
where \(P_{m_{i}},P_{e_{i}},H_{i}\), and \(w_{i}\) are the mechanical power of the turbine in p.u., the electrical power in p.u., the inertia of \(i\)th power plant in seconds, and the speed (frequency) in p.u. of generator \(i\), respectively. Interconnected power systems consist of several generating units connected to loads by an electric grid. Practically, to reduce the calculation burden and simplify the complexity of interconnected power system models, it is better to equivalent the different generation units on a single aggregated unit [35; 36]. To this end, the equivalent quantities can be determined based on (4), and dynamic behavior of the aggregated system is represented by (5) as follows  
\[ \begin{array}{l}H_{COI}=\frac{\sum\limits_{i=1}^{N}H_{i}MVA_{i}}{MVA_{base}} \\ P_{m}=\sum\limits_{i=1}^{N}P_{mi}\\ P_{e}=P_{loss}+\sum\limits_{i=1}^{N}P_{li}\\ w=\frac{\sum\limits_{i=1}^{N}H_{i}w_{i}}{\sum\limits_{i}^{N}H_{i}}\\ \end{array}  \tag{4}\]  
\[\frac{dw}{dt}=\frac{1}{2H_{COI}}\left(P_{m}-P_{e}\right) \tag{5}\]  
where \(P_{m},P_{e},P_{loss},P_{li},H_{COI}\), \(w_{i}\), and \(\Delta w_{ref}\) are the mechanical power of the equivalent turbine in p.u., the electrical demand in p.u., active power losses in the electric network in p.u., electrical active powers of loads in p.u., the averaged center of inertia of the system in seconds, the average system frequency, and the frequency set of each generating unit, respectively. Governor-turbine system plays a vital role in converting the primary energy into mechanical energy which is converted to electric energy by synchronous generators. The dynamic model of this system has great impacts on the power SFR during normal and emergency operation situations. In last decade, many models with different orders are suggested for governor-turbine systems. However, it has been proven in [35; 36; 37] that the frequency response model based on the swing equation dynamics is sufficient for frequency studies. The dynamics of these important components are represented as their transfer functions as shown in Figure 1.  
For a power system which consists mostly of steam boiler power plants, the frequency response of the system in the event of a disturbance can be represented by a simple frequency response model [37]. The model consists generally of an equivalent governor system regulation, system inertia, and system load damping. For a system with more than one generating units, each unit is modeled separately in the SFR model. An SFR model which consists of \(N\) generators is of \(N+1\) dynamic order. Figure 1 shows the general layout of the SFR model which includes the UFLS scheme. The frequency response dynamic behavior of the system in the event of a disturbance is given by Equation (6). For simplicity, the equation can be expressed in partial form as given by Equation (7). It can be noted that in real systems, frequency is a time varying parameter therefore by Laplace conversions, Equation (8) gives the time response of the system.