## 2 Materials and Methods
### Control Analysis for a Single Machine Infinite Bus System
#### 2.2.2 A Trajectory Tracking Approach for SMIB  
Without loss of generality, a single machine infinite bus power system is first considered in this section. A new output feedback adaptive neural control strategy based on differential flatness for both transient stabilization and robust tracking of desired power-angle reference profiles for a synchronous generator in interconnected power systems is proposed. Trajectory planning considers that transient stabilization of the system at a specified equilibrium (operation) point in the state space should be firstly performed. Next, the power system should be efficiently transferred to another desired operation condition by following a desired smooth reference profile into the designed safe operational region of the power system. The controlled nonlinear dynamic system model (5) exhibits the structural property of differential flatness. That is, state and control variables can be parameterized in terms of a so-called flat output and a finite number of its time derivatives [25; 26]. In fact, the power angle \(y=\delta\) constitutes a flat output. From the time derivatives up to the third order of the flat output \(y\), the differential parametrization of the system variables results in  
\[\delta= y\] \[\omega= \dot{y}\] \[E= f(y,\dot{y},\dot{y})\] \[u= \frac{1}{\gamma}\Big{[}\ddot{y}+\gamma aE-\gamma bY\cos(y+\theta) -\gamma E_{f}+\sigma YE\dot{y}\cos(y+\theta)\] \[+\sigma D\ddot{y}-\sigma\dot{P}_{m}\big{]} \tag{7}\]  
with control input gain \(\gamma\) given by  
\[\gamma=-\sigma Y\sin(y+\theta)-2\sigma GE \tag{8}\]  
Figure 1: Single machine infinite bus power system. Hence, the flat output dynamics can be described with the following nonlinear differential equation:  
\[\ddot{\psi}+\gamma aE-\gamma bY\cos(y+\theta)-\gamma E_{f}+\sigma YE\dot{y}\cos(y +\theta)+\sigma D\ddot{y}=\gamma u-\sigma\dot{P}_{m} \tag{9}\]  
An output feedback tracking controller for power-angle reference profiles \(y^{*}(t)=\delta^{*}(t)\) can be then synthesized as  
\[u= \frac{1}{\gamma}\bigg{[}\ddot{y}^{*}-\beta_{3}(\ddot{y}-\ddot{y}^{*})- \beta_{2}(\dot{y}-\dot{y}^{*})-\beta_{1}(y-y^{*})-\beta_{0}\int_{0}^{t}(y-y^{* })dt-\xi\bigg{]} \tag{10}\]  
with  
\[\xi= \gamma\Big{[}-aE+bY\cos(y+\theta)+E_{f}\Big{]}-\sigma YE\dot{y} \cos(y+\theta)\] \[\omega= \dot{y}\]0  
In the present study, power-angle reference trajectories \(y^{*}(t)\) to perform a smooth transference of the power system from an operating condition to another are described via Bezier curves [27]. The time derivatives of planned reference trajectories are then described via closed form mathematical expressions. Differentiation with respect to time of output signals can be implemented as well [28]. The integral reconstruction approach of time derivatives of output signals constitutes another alternative to be considered in the control design problem [29]. Implementation of the controller (10) into the controlled power system yields the closed-loop tracking error dynamics, \(e_{\delta}=y-y^{*}\),  
\[\omega= \dot{y}\]1  
Therefore, asymptotic tracking of the reference trajectory \(y^{*}(t)=\delta^{*}(t)\) can be guaranteed by selecting the control design parameters \(\beta_{j}\), \(j=0,\ldots,3\), so that the characteristic polynomial associated with Equation (12) is a Hurwitz polynomial. Then,  
\[\omega= \dot{y}\]2  
In this context, the following family of stable polynomials can be used to compute the controller parameters \(\beta_{j}\):  
\[\omega= \dot{y}\]3  
with \(p>0\). Then, the parameters \(\beta_{j}\), \(j=0,\ldots,3\), can be algebraically calculated in terms of a positive constant \(p\) as follows  
\[\omega= \dot{y}\]4 \[\omega= \dot{y}\]5 \[\omega= \dot{y}\]6 \[\omega= \dot{y}\]7  
Certainly, the differential flatness controller (10) stands as a very good alternative for the tasks of stabilization and tracking of desired power-angle reference profiles for operational scenarios where the system parameters and nonlinear dynamics are known accurately.