## 3 Design of the Proposed Controller for the Power System
### Structure of the Proposed Controller  
The FOPD-FOPID cascaded controller has been intricately designed to enhance the stability of frequency within power systems that incorporate wind turbines. Its primary objective is to minimize deviations in frequency caused by load disturbances, disruptions from wind turbines, and uncertainties in power system parameters. This elaborate controller consists of two components: the FOPD controller and the FOPID controller. The FOPD controller primarily determines the outcome, while the FOPID controller shapes and guides it by adjusting the reference signal. In this proposed control framework, the FOPD controller is referred to as the main, primary, or external controller, and the FOPID controller is referred to as the internal, secondary, or sub-controller. These components work synergistically, as depicted in Figure 3, to establish cohesive coordination between the LFC system and the SMES. Figure 3 illustrates the suggested setup of the cascaded controller, which facilitates the orchestration of the inner loop dynamics through Equation (3). Equation (3) succinctly represents the transfer function that exclusively encapsulates the inner loop dynamics, as demonstrated in Figure 3:  
\[Y_{2}(s)=M_{2}(s)U_{2}(s) \tag{3}\]  
In Equation (3), \(M_{2}\)(s) embodies the transfer function of the internal process, whereas \(U_{2}\)(s) signifies the input signal directed towards said process. The principal controller (FOPD), situated in the outer layer, adeptly curtails any oscillations pertaining to frequency and mitigates external interferences. Equation (4) reveals the transfer function characterizing the outer loop's operation:  
\[Y(s)=M_{1}(s)U_{1}(s) \tag{4}\]  
where \(M_{1}\)(s) symbolizes the transfer function of the external process, while \(U_{1}\)(s) stands for the input signal directed towards said process. As Figure 3 illustrates, \(N_{2}\)(s) denotes the FOPID controller found in the inner layer, whereas \(N_{1}\)(s) represents the FOPD controller situated in the outer layer. The transfer functions of both FOPD and FOPID controllers are demonstrated through Equations (5) and (6):  
\[N_{1}(s)=K_{P1}+K_{d1}s^{\mu_{1}} \tag{5}\]  
\[N_{2}(s)=K_{P2}+K_{I}s^{-\lambda}+K_{d2}s^{\mu_{2}} \tag{6}\]  
Figure 4 shows the inside structure of the FOPD-FOPID controller.  
The settings of the suggested controller are found by making the _ISE_ objective function as small as possible by using the DOSA method. Equation (7) shows the main goal of _ISE_. The limits or boundaries of the objective functions are represented by Equation (8):  
\[ISE=\int\limits_{0}^{t_{s}}{(\Delta f)^{2}dt} \tag{7}\]\[\begin{array}{l}K_{P1,\min}\leq K_{P1}\leq K_{P1,\max}\\ K_{d1,\min}\leq K_{d1}\leq K_{d1,\max}\\ \mu_{1,\min}\leq\mu_{1}\leq\mu_{1,\max}\\ K_{P2,\min}\leq K_{P2}\leq K_{P2,\max}\\ K_{I,\min}\leq K_{I}\leq K_{I,\max}\\ K_{d2,\min}\leq K_{d2}\leq K_{d2,\max}\\ \lambda_{\min}\leq\lambda\leq\lambda_{\max}\\ \mu_{2,\min}\leq\mu_{2}\leq\mu_{2,\max}\\ \end{array} \tag{8}\]  
Figure 3: The suggested setup of a cascaded controller manifests itself in order to orchestrate cohesive coordination between the LFC system alongside SMES.