## 3 Materials and Methods
When \(A(q)\), \(C(q)\), and \(D(q)\) are equal to 1, then the expression for the GL polynomial model is written as the output-error (OE) model:  
\[y(n)=\frac{q^{-k}B(q)}{F(q)}u(n)+e(n)=\frac{B(q)}{F(q)}u(n-k)+e(n) \tag{4}\]  
The signal scheme of the OE model is shown in Figure 2.  
The polynomial functions \(B(q)\) and \(F(q)\) for \(q^{-1}\) can be expressed as:  
\[B(q)=b_{0}+b_{1}q^{-1}+\cdots+b_{n}q^{-nb+1}, \tag{5}\]  
\[F(q)=1+f_{1}q^{-1}+\cdots+f_{nf}q^{-nf}. \tag{6}\]  
The OE model is a polynomial model with simpler configuration that consists of two active polynomials--\(B\) and \(F\). This model is a traditional transmission function that associates the measured input with output while considering white noise as an additional output interference. The OE model uses time and frequency domain data. This model describes the system dynamics without the use of interference characteristic modeling parameters. Other parametric identification models such as the estimated BJ and ARMAX can also be used. The following is the methodology for the development of the SI algorithm:  
1. First, the mathematical model of the traditional synchronous generator is modified to extract the effect of inertia in the event of interference in the electrical power system as accurately as possible. For this purpose, the generator power system stabilizer and the governor model system are deactivated. The required inertia is also determined in the generator model. 2. Disturbance is modeled/simulated in the electrical power system by disconnecting the generating sources so that the frequency change differs several times in the electrical power system compared with the calculation errors (e.g., 300 mHz). It is essential that the electrical power system remains stable after the disturbance and that the transient process reaches a steady state. 3. The frequency change in the electric power system at the point of common coupling (PCC) and the change in the active power of the generator is recorded. The following transmission functions are created:  
Figure 1: The schematic diagram of a generalized typical discrete-time linear mathematical model [34]. Figure 2: Signal flow of OE model. \[B(q) =P_{G}(t), \tag{7}\] \[F(q) =f_{G}(t). \tag{8}\]
4. Using the OE model of parametric identification and using Functions (7) and (8) obtained in the 3 rd methodology point, the SI algorithm is generated with the input signal: \[H=\frac{1}{2}\frac{\textit{J}\omega_{0m}^{2}\times 10^{-6}}{\textit{MVA rating}}. \tag{3}\]0 (9) \[H=\frac{1}{2}\frac{\textit{J}\omega_{0m}^{2}\times 10^{-6}}{\textit{MVA rating}}. \tag{3}\]1 (10) The second-order transmission function of the SI algorithm is as follows (11): \[H=\frac{1}{2}\frac{\textit{J}\omega_{0m}^{2}\times 10^{-6}}{\textit{MVA rating}}. \tag{3}\]2 (11)
5. The error of the output signal \(y(n)\) of the SI algorithm with the output power of the generator \(P_{G}(t)\) is determined. \(R_{y\hat{y}}\), RMSE, and R\({}^{2}\) error estimation methods are used to determine the accuracy of the algorithm. The Transmission System Operator (TSO) assesses the frequency stability of the power system according to the RoCoF value and determines how many additional SI algorithms need to be implemented to ensure the stability of the electrical power system. The flowchart provided in Figure 3 provides the step-by-step SI algorithm methodology. \[H=\frac{1}{2}\frac{\textit{J}\omega_{0m}^{2}\times 10^{-6}}{\textit{MVA rating}}. \tag{3}\]3 Flowchart of SI algorithm methodology.