## 2 Mathematical Modelling of Investigated Power System
### Design of the Fractional Order Fuzzy Pre-Compensated Intelligent PID (FOFP-iPID) Controller  
A novel hybrid fractional order fuzzy pre-compensated intelligent PID (FOFP-iPID) controller is proposed in this study, where the suggested controller is synthesised from the fractional order fuzzy pre-compensator (FOFP) and the fractional order intelligent PID (FOiPID) controller. Firstly, the FOFP controller is subjected to modification of the control signal for compensating overshoots/undershoots in transient output response and also amended the dynamic performances in the presence of parameter variations and load perturbations. Then, the FOiPID controller is equipped to make the controller more robust, effective, and to ameliorate the dynamic performance. As a result, the suggested FOFP-iPID controller is comprised of unique features of the fuzzy pre-compensated controller with FO, which aimed to overcome the problem of LPs in the output response. The advantage of the suggested FOFP-iPID controller is simply in its architecture, high efficiency and is the first time it has been employed for LFC strategy. The proposed FOFP-iPID controller is highlighted in Figure 2. In Figure 2, the dynamics of the proposed hybrid fractional order fuzzy pre-compensated intelligent PID (FOFP-iPID) controller can be described by the following equations. From Figure 2 the area control error (\(e_{i}(t)=ACE_{i}\)) can be formulated as:  
\[e_{i}(t)=B_{i}\Delta f_{i}+\Delta P_{tieij} \tag{20}\]  
From Figure 2:  
\[u_{1}(t)=K_{pi}\frac{d^{-\eta}u_{FLCi}(t)}{dt^{-\eta}}\left(K_{e1}e_{i}(t),K_ {e2}\frac{d^{\delta}e_{i}(t)}{dt^{\delta}}\right) \tag{21}\]  
\[u_{2}(t)=K_{pd}u_{FLCi}\left(K_{e1}e_{i}(t),K_{e2}\frac{d^{\delta}e_{i}(t)}{dt^ {\delta}}\right) \tag{22}\]  
where, \(\eta\) and \(\delta\) signify the order of integro-differentiator of the FOFP controller, \(K_{e1}\) and \(K_{e2}\) represent the input gains, and \(u_{FLCi}\) is the fuzzy output signal. Now, combining Equations (21) and (22) gets the controller signal (\(u_{FOFPIDi}(t)\)), in which \(u_{FOFPIDi}(t)\) represents the pre-compensated term of the proposed strategy. \[u_{FOFPIDi}(t)=K_{pi}\frac{d^{-\eta}u_{FLCi}(t)}{dt^{-\eta}}\left(K_{e1}e_{i}(t ),K_{e2}\frac{d^{\delta}e_{i}(t)}{dt^{\delta}}\right)+K_{pd}u_{FLCi}\left(K_{e1 }e_{i}(t),K_{e2}\frac{d^{\delta}e_{i}(t)}{dt^{\delta}}\right) \tag{23}\]  
Now, the new error \(e_{i}^{u}(t)\) which is incorporated into the fractional order intelligent (FOiPID) controller can be derived from Figure 2:  
\[e_{i}^{u}(t)=u_{FOFPIDi}(t)+ACE_{di} \tag{24}\]where, \(ACE_{di}\) denotes the desired error of \(ACE_{i}\). Finally, the controller signal \((u_{i}(t))\) of the proposed FOFP-iPID controller can be derived directly from Figure 2 by using the fractional order intelligent PID controller concept:  
\[u_{i}(t)=\frac{1}{\bar{\alpha}}\left(-\hat{e}(t)+y^{(n)*}+K_{p}e_{i}^{u}(t)+K_{i} \frac{d^{-\lambda}e_{i}^{u}(t)}{dt^{-\lambda}}+K_{d}\frac{d^{\mu}e_{i}^{u}(t)}{ dt^{\mu}}\right) \tag{25}\]  
As shown in Figure 2, the terms \(K_{e1}\), \(K_{e2}\), \(K_{pi}\) and \(K_{pd}\) are the scaling factors (SFs) for the fractional order fuzzy logic pre-compensated (FOFP) controller. Furthermore, \(\lambda\), \(\mu\), \(\eta\) and \(\delta\) represent the extra fractional order parameters, which can be used in the control system that will increase the flexibility of regulating the parameters. In doing so, the associated gains and the scaling factors (SFs) are optimized by employing the ALO algorithm. The implementation phases of the FLC inference are mainly comprised of the fuzzy rule base, membership functions, fuzzification and defuzzification processes.