# Optimal Non-Integer Sliding Mode Control for Frequency Regulation in Stand-Alone Modern Power Grids
## 3 Description of an Isolated Fractional-Order Microgrid Model
### Structure of the LFC-Based MG System  
Figure 3 illustrates the structure of the suggested LFC for the test MG, which employs different DGs such as PV, WT and DEG, storage devices (BESS and FESS), and loads. It is clear that the PV, fuel cell (FC), and BESS parts are connected to the AC MG via DC/AC interfacing inverters. All small-scale DGs and energy storage sections are connected to the AC bus via a circuit breaker. The spinning reserve for the secondary frequency control is offered by the diesel power system.  
The dynamics of the MG system is shown in Figure 3. This system is a nine-state set of fractional-order equations. The objects of state equations are presented in the following:  
\[\begin{array}{l}D^{\beta}(\Delta f)=\frac{1}{2H}\Big{[}T_{s_{flat}}+T_{md}+T _{w}-\Delta P_{L}+T_{f_{flat}}-T_{bat}-D*\Delta f\Big{]}\\ D^{\beta}(T_{s_{inv}})=\frac{1}{T_{inv}}\Big{[}T_{s}-T_{s_{inv}}\Big{]}\\ D^{\beta}(T_{s_{fill}})=\frac{1}{T_{fill}}\Big{[}T_{s_{inv}}-T_{s_{fill}} \Big{]}\\ D^{\beta}(T_{gd})=\frac{1}{T_{g}}\Big{[}T_{cd}-\frac{\Delta f}{R}-T_{gd}\Big{]} \\ D^{\beta}(T_{md})=\frac{1}{T_{d}}\Big{[}T_{gd}-T_{md}\Big{]}\\ D^{\beta}(T_{fc})=\frac{1}{T_{fc}}\Big{[}T_{cf}-\frac{\Delta f}{R}-T_{fc} \Big{]}\\ D^{\beta}(T_{f_{inv}})=\frac{1}{T_{inv}}\Big{[}T_{fc}-T_{f_{inv}}\Big{]}\\ D^{\beta}(T_{f_{fit}})=\frac{1}{T_{filt}}\Big{[}T_{f_{inv}}-T_{f_{fit}}\Big{]} \\ D^{\beta}(T_{bat})=\frac{1}{T_{b}}\big{[}\Delta f-T_{bat}\big{]}\end{array} \tag{11}\]  
Figure 2: Model of the diesel engine generator (DEG) system.  
The nine state MG dynamics and the output equation rewrite in a compact form in the following equations:  
\[\begin{array}{c}D^{\alpha}X=AX+BU\\ Y=CX+DU\end{array} \tag{12}\]  
where \(X,A,B,C,D\), and \(U\) are stated matrix, system matrix, input matrix, the direct transition matrix, and input respectively, and they are introduced as follows:  
\[X=\begin{bmatrix}\Delta f\ T_{s_{inv}}\ T_{s_{flat}}\ T_{gd}\ T_{md}\ T_{fc}\ T_{f_{inv}}\ T_{f_{flat}}\ T_{bat}\end{bmatrix}^{T}\]  
\[A=\begin{bmatrix}-\frac{D}{2H}&0&\frac{1}{2H}&0&\frac{1}{2H}&0&0&\frac{1}{2H} &-\frac{1}{2H}\\ 0&-\frac{1}{i_{inv}}&0&0&0&0&0&0\\ 0&\frac{1}{i_{fin}}&-\frac{1}{i_{fin}}&0&0&0&0&0\\ -\frac{1}{Rt_{g}}&0&0&\frac{1}{t_{g}}&0&0&0&0\\ 0&0&0&\frac{1}{t_{T}}&-\frac{1}{t_{T}}&0&0&0&0\\ \frac{1}{Rt_{fc}}&0&0&0&-\frac{1}{t_{fc}}&0&0&0\\ 0&0&0&0&\frac{1}{i_{inv}}&-\frac{1}{i_{inv}}&0&0\\ 0&0&0&0&0&\frac{1}{i_{fin}}&-\frac{1}{i_{fin}}&0\\ \frac{1}{t_{b}}&0&0&0&0&0&0&-\frac{1}{2H}\end{bmatrix}\]  
Figure 3: The overall microgrid (MG) scheme for load frequency control (LFC). FOSMC: fractional order sliding mode control.  
\[B=\left[\begin{array}{ccc}0&0\\ 0&0\\ 0&0\\ 1/t_{g}&0\\ 0&0\\ 0&1/t_{fc}\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ \end{array}\right],\ N=\left[\begin{array}{ccc}-1/2H&1/2H&0\\ 0&0&1/t_{inv}\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ \end{array}\right],\]  
\[U=\left[\begin{array}{ccc}T_{cd}&T_{cf}\\ \end{array}\right]^{T},\ Q=\left[\Delta P_{L}\ T_{w}\ T_{s}\right]^{T}\]  
\[C=\left[\begin{array}{ccc}1&0&0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{array}\right].\]  
Table 1 describes the parameters of the MG equations utilized in the simulation.