## 4 Proposed Optimal UFLS Based on WAMS  
Using WAMS, the frequency behavior can be easily predicted since measurements are constantly being monitored in real time, therefore, it is possible to predict the future few-seconds response of the system after a disturbance using the realized SFR based on WAMS as shown in (6)-(8). For example,  
Figure 1: the general layout of the system frequency response (SFR) model. we can online calculate the time when the derivative of (8) is equal to zero, then by substituting the calculated time in (8), we can predict the minimum frequency that the system will reach. Furthermore, the future frequency behavior can be easily predicted using (8) at any time instance. To implement WAMS-based optimal multi-stage UFLS, we assume that the disturbance is online estimated using the method proposed in [3]. Firstly, a lumped equivalent model, which represents the dynamics of the whole power system, is developed online. From the equivalent model, the center of inertia is obtained which can be used in the swing equation to calculate the overall disturbance in the power system. Instead of monitoring the frequency of different generating units, the overall center of inertia frequency is monitored which is used to online estimate the disturbance with better accuracy. Based on the estimated magnitude of disturbance, the systems' frequency nadir, new steady-state frequency and the rate of change of frequency (ROCOF) can be accurately obtained. Finally, the most optimum amount of load to be shed is determined and UFLS can be implemented to restore the frequency to permissible level. By combining the UFLS signal in the SFR model, the state space model of the system can be expressed as follows  
\[\frac{dx}{dt}=Ax+B\left(\Delta P_{0}-U\left(t\right)\right) \tag{9}\]  
From Equation (9), \(A\) and \(B\) are matrices describing the system response where a disturbance \(\Delta P_{0}\) has occurred. The amount of load to be shed for mitigating the effects of the disturbance such that the frequency will be restored to permissible range is given by \(U(t)\). When a disturbance occurs, the event can be a generator loss or line tripping, the dynamics of the power system are observed from the modified swing equation which includes the load-shed to keep the frequency within a desired level. At time instance \(\left(t=0\right)\), values needed for online model realization such as \(P_{m}\) are provided by WAMS and at this point \(\Delta w\left(0\right)=0\). When disturbance occurs, the total load-shedding amount which is as an input to the system described by Equation (9) needs to be optimized such that the frequency is restored to the predefined range. The optimization model has the following basic constraints which must be satisfied in the design of the UFLS: \(i)\) The mechanical power \(\left(P_{m}\right)\) delivered by the turbine must not exceed certain limits; \(ii)\) The steady-state frequency \(\left(f_{ss}\right)\) should be higher than a set threshold limit at any given time; \(iii)\) The minimum time \(\left(t_{m}\right)\) required to restore the frequency to a certain minimum acceptable operation value should be lower than a predefined time limit; \(iv)\) The maximum allowable number of load-shedding stages \(\left(k\right)\); \(v)\) The minimum and maximum amount of load-shed for each stage \(\left(P_{stage}\right)\); \(vi)\) The total load-shed \(\left(P_{shed}\right)\) allowable as compared to the magnitude of the disturbance; \(vii)\) The minimum allowed frequency \(\left(f_{tss}\right)\); \(viii)\) minimum allowable frequency bands or the minimum margin \(\left(\sigma\right)\) between two consecutive frequency thresholds; and finally \(ix)\) allowable time delay limits \(\left(t_{d}\right)\). With the constraints stated above, the proposed optimization problem is generally described as follows  
\[ \begin{array}{l}\min F_{obj}(Z)\\ s.t\\ LB\leq Z\leq UB\end{array}  \tag{10}\]  
\(F_{obj}(Z)\) if the adopted objective function.