# Optimal Non-Integer Sliding Mode Control for Frequency Regulation in Stand-Alone Modern Power Grids
## 4 Proposed Fractional-Order Sliding Mode Control Scheme  
Here, a non-integer sliding surface is designed as follows:  
\[S(t)=X+D^{-\alpha}[\ K(|X|^{p})sign(X)] \tag{13}\]  
where \(1<p<2\) and \(K\), is a vector of positive constants.  
Based on the sliding mode control theory, when the system operates in the sliding mode, it should satisfy:  
\[S(t)=0 \tag{14}\]  
So, using property \(1\), \(D^{\alpha}S(t)=0\). Therefore:  
\[D^{\alpha}S(t)=D^{\alpha}X+K(|X|^{p})sign(X)=0 \tag{15}\]  
The above equation can be rewritten in the form:  
\[D^{\alpha}X(t)=-K(|X|^{p})sign(X) \tag{16}\]  
Based on the frequency distributed model theorem, this sliding mode dynamic is stable, and its state trajectories converge to the equilibrium \(X=0\).  
As a proof, due to Theorem 2, FO sliding dynamics (Equation (16)) can be expressed as:  
\[\left\{\begin{array}{c}\frac{\partial\Lambda(\omega,t)}{\partial t}=-\alpha ^{2}\Lambda(\omega,t)-K|X|^{p}sgn(X)\\ X(t)=\int_{0}^{\infty}u(\omega)\Lambda(\omega,t)d\omega\end{array}\right. \tag{17}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Symbol and Abbreviation** & **Values** & **Symbol and Abbreviation** & **Values** \\ \hline \(\Delta P_{L}\) (change in load power) & 0.02 s & \(T_{g}\) (governor time constant) & 0.08 s \\ \(T_{inv}\) (time constant of inverter circuit of solar unit) & 0.04 s & \(T_{b}\) (battery power time constant) & 0.1 s \\ \(T_{fc}\) (time constant of fuel cell) & 0.26 s & 2H (inertia constant) & 0.1667 \\ \(T_{filt}\) (time constant of filter circuit of solar unit) & 0.004 s & D (damping coefficient) & 0.015 \\ \(T_{t}\) (diesel generator time constant) & 2.00 s & R (DG speed regulation) & 3 \\ \hline \end{tabular}
\end{table}
Table 1: The MG power system’s parameters.  
By choosing a positive Lyapunov function in the form \(V_{1}=\frac{1}{2}\int_{0}^{\infty}u(\omega)\Lambda^{2}(\omega,t)d\omega\), one has:  
\[\begin{array}{l}D^{\alpha}V_{1}=\int_{0}^{\infty}u(\omega)\Lambda(\omega,t) \frac{\partial\Lambda(\omega,t)}{\partial t}d\omega\\ =\int_{0}^{\infty}u(\omega)\Lambda(\omega,t)\Big{[}-\omega^{2}\Lambda(\omega,t )-K|X|^{p}sgn(X)\Big{]}d\omega\\ =-\int_{0}^{\infty}u(\omega)\omega^{2}\Lambda^{2}(\omega,t)d\omega-K|X|^{p}sgn (X)\int_{0}^{\infty}u(\omega)\Lambda(\omega,t)d\omega\\ =-\int_{0}^{\infty}u(\omega)\omega^{2}\Lambda^{2}(\omega,t)d\omega-K|X|^{1+p}< 0\end{array} \tag{18}\]  
So, based on Theorem 1, the FO sliding dynamics (Equation (16)) is asymptotically stable.  
Now, to guarantee the existence of the sliding motion, the following robust controller is designed:  
\[U(t)=-\Big{[}B^{-1}AX+KB^{-1}(|X|^{p})sign(X)+\gamma\;B^{-1}\;sign(S)\;+ \lambda B^{-1}\;|S|\tan\mathrm{h}(S)\Big{]} \tag{19}\]  
In which \(\gamma,\lambda\) are vectors of positive constants.  
**Theorem 3**: _Consider the fractional order linear system (12). If this system is controlled by the control law (19), then the state trajectories of the system will converge to the equilibrium point._  
**Proof.** Choose the following Lyapunov function  
\[V_{2}(t)=|S| \tag{20}\]  
Applying the \(D^{\alpha}\) of \(V_{2}(t)\) and using Lemma 1, one obtains  
\[D^{\alpha}V_{2}(t)=D^{\alpha}|S|=sign(S)D^{\alpha}S \tag{21}\]  
Substituting \(s_{i}\) from (13) into (21), one has  
\[D^{\alpha}V_{2}(t)=sign(S)[D^{\alpha}X+K(|X|^{p})sign(X)] \tag{22}\]  
Inserting \(D^{\alpha}X\) from (12), we have  
\[\begin{array}{l}D^{\alpha}V_{2}(t)=sign(S)[AX+BU+K(|X|^{p})sign(X)]\\ =sign(S)[AX+B\Big{\{}-\Big{[}B^{-1}AX+KB^{-1}(|X|^{p})sign(X)+\gamma B^{-1} sign(S) \tag{23}\]  
Now some simplifications, and based on Lemma 1, we obtain  
Case 1: If \(S>0\), then \(sign(S)=1\) and \(0<\tan\mathrm{h}(S)=\varepsilon<1\), then  
\[D^{\alpha}V_{2}(t)=-[\gamma+\lambda|S|\varepsilon]<0 \tag{24}\]  
Case 2: If \(S<0\), then \(sign(S)=-1\) and \(-1<\tan\mathrm{h}(S)=\xi<0\), thus  
\[D^{\alpha}V_{2}(t)=[-\gamma+\lambda|S|\xi]<0 \tag{25}\]  
Thus, according to Theorem 1, the state trajectories of the fractional order system (11) will converge to \(s_{i}=0\) asymptotically. \(\Box\)