## 2 Mathematical Modelling of Investigated Power System
### Design of the Fopid Controller  
Fractional order (FO) is a generalization of integer order integral and differential to the FO operator \({}_{a}D_{t}^{a}\) where \(a\) and \(t\) signify the limits of the operation and \(\alpha\) is the FO which is a complex number. Thus, the FO operator can be stated as:  
\[{}_{a}D_{t}^{a}=\left\{ \begin{array}{cc}d^{a}/dt^{\alpha},&\Re\left(\alpha \right)>0\\ 1,&\Re\left(\alpha\right)=0\\ \int_{a}^{t}\left(d\tau\right)^{\alpha},&\Re\left(\alpha\right)<0\end{array} \right. \tag{10}\] \begin{table}
\begin{tabular}{l l l l} \hline \(T_{gs}\) & Governor time constant of steam turbine & \(B_{g}\) & Time constant of the valve positioner \\ \(T_{t}\) & Steam turbine time constant & \(T_{gb}^{\alpha}\) & Hydro turbine governor time constant \\ \(T_{t}\) & Steam turbine reheat time constant & \(T_{id}^{\alpha}\) & Compressor discharge volume time constant \\ \(K_{g}\) & Steam turbine reheat constant & \(C_{g}^{\alpha}\) & Gas turbine valve positioner \\ \(X_{g}\) & Lead time constant of gas turbine governor & \(T_{f}^{\alpha}\) & Gas turbine fuel time constant \\ \(Y_{t}^{\alpha}\) & Lag time constant of gas turbine governor & \(T_{g}^{\alpha}\) & Gas turbine combustion reaction time delay \\ \(T_{n}\) & Starting time of water in hydro turbine & \(T_{n}\) & Hydro turbine speed governor reset time \\ \(T_{p}\) & Power system time constants & \(PF_{Th}\) & Participation factors of thermal unit \\ \(PF_{Hy}\) & Participation factors of hydro unit & \(PF_{G}\) & Participation factors of gas unit \\ \(K_{g}\) & Power system gains & \(T_{12}\) & Synchronizing coefficient \\ \(R_{Th}\) & Governor speed regulation parameters of thermal unit & \(R_{Hy}\) & Governor speed regulation parameters of hydro unit \\ \(R_{G}\) & Governor speed regulation parameters of gas unit & \(B_{1}\) & Frequency bias coefficients of area-1 \\ \(B_{2}\) & Frequency bias coefficients of area-2 & & \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters description of the proposed system. There are different descriptions and approximations for FO derivative and integral [17]. The FO derivative based on the Riemann-Liouville (R-L) definition is stated by:  
\[{}_{a}D_{t}^{a}f(t)=\frac{1}{\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}}\int_{a}^{t} \left(t-\tau\right)^{n-\alpha-1}f(\tau)d\tau \tag{11}\]  
where \(n-1>\alpha\geq n\). The FO integral based on R-L definition modeled by:  
\[{}_{a}D_{t}^{a}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\left(t-\tau\right)^{ \alpha-1}f(\tau)d\tau \tag{12}\]  
For simplicity, Laplace transformation (LT) is routinely employed to designate the fractional order. In general \(\left(0~{}<~{}\alpha~{}<~{}1\right)\), and the LT of the R-L fractional derivative or integral is modeled by:  
\[L\left\{{}_{a}D_{t}^{a}f(t)\right\}=s^{a}F(s)-\sum_{k=0}^{n-1}s^{k}{}_{a}D_{t }^{a-k-1}f(t)|_{t=0} \tag{13}\]  
Under zero initial conditions for  
\[L\left\{{}_{a}D_{t}^{\pm a}f(t)\right\}=s^{\pm a}F(s) \tag{14}\]  
This signifies that when zero initial concepts are concerned, the dynamic responses of a plant designated by FO differential equations turns to transfer functions with fractional orders of's' (Laplace operator) [20]. The best performance of the FOs relies on the order of the estimation (\(\bar{N}\)). Minimum values of \(\bar{N}\) may drive ripples in the phase behaviors. These waves can be handled by raising the values of \(\bar{N}\), but the computational load of the estimation becomes difficult and complex in the hardware implementation [21]. In this article, 5\(th\) order [PERSON]'s recursive estimation is considered while fitting frequency variety \([\omega_{l},~{}\omega_{h}]\) is taken as [0.01, 100]\(rad/s\). The architecture of the FO designed in this study is performed with employed MATLAB/Simulink based toolboxes [22]. The purpose of the toolbox is to furnish a user-friendly, rapid and precise tool for simulating different FO controllers for several practices.