## 2 System Layout  
The total flow of energy delivered from any number of sources in a MG to the end user can be modelled as a single current controlled source of a setpoint (I\({}^{*}_{\text{ref}}\)) connected to a parallel RC branch, through the principle of superposition, a single virtual source can be constructed in Figure 1.  
There are two key dynamics of interest within the system. The first is of the output voltage (V\({}_{\text{out}}\)) in response to the input current (I\({}_{\text{source}}\)), as the derivative of the bus voltage is proportional to the imbalance between the supply current and the load current (I\({}_{\text{load}}\)). \[\frac{\text{dV}_{\text{out}}}{\text{dt}}=\frac{1}{C}(\text{I}_{\text{source} }-\text{I}_{\text{load}}) \tag{1}\]  
\[\text{I}_{\text{load}}=\frac{\text{V}_{\text{out}}}{\text{R}_{\text{load}}} \tag{2}\]  
This subsequently leads to the transfer function relating the output voltage and the input current where R\({}_{\text{bat}}\) is the load's resistance  
\[\frac{\text{V}_{\text{out}}}{\text{I}_{\text{source}}}=\frac{\text{R}_{\text {load}}}{(\text{R}_{\text{load}}\text{C}\cdot\text{s}+1)} \tag{3}\]  
The second dynamic of interest is that of the input current in response to the input current reference (I\({}_{\text{ref}}\)). There are a multitude of apparatus that can achieve current control mode of an electric source, regardless, as with any real system, it is quite reasonable to assume that such systems are quantifiable, usually using a nonlinear second order system to encapsulate common characteristics such as natural frequency (\(\omega_{\text{n}}\)), Damping coefficient (\(\zeta\)), settling time, overshoot, steady-state error, etc. \[\frac{\text{I}_{\text{s}}}{\text{I}_{\text{s}}^{\text{ref}}}=\frac{\omega_{ \text{n}}^{2}}{\text{s}^{2}+2\zeta\omega_{\text{n}}\cdot\text{s}+\omega_{ \text{n}}^{2}} \tag{4}\]  
To validate this assumption a simple boost converter is deployed as shown in Figure 2.  
Figure 1: Microgrid single current sources model. The control signal \"u\" dictates the conduction state of the MOSFET, a value of 1 or a logical high corresponded to the conduction state whereas a value of 0 or logical low corresponded to the off state as shown in Figure 3. The symbol \(\
eg\) indicates the NOT operator. A state space model may be adopted to describes its behaviour in time where lower-case letter i denotes the input current to the converter, E, L and C are the input voltage, inductance and capacitor respectively. \[ \begin{bmatrix}\frac{\mathrm{d}\mathrm{i}}{\mathrm{d}\mathrm{f}}\\ \frac{\mathrm{d}^{\prime}\mathrm{f}_{\mathrm{out}}}{\mathrm{d}\mathrm{t}} \end{bmatrix} = \begin{bmatrix}0&-\frac{-\mathrm{u}\mathrm{i}}{\mathrm{i}}\\ \frac{-\mathrm{u}\mathrm{i}}{\mathrm{i}}&-\frac{\mathrm{i}}{\mathrm{i}_{ \mathrm{load}}\mathrm{C}}\end{bmatrix}\begin{bmatrix}\mathrm{i}\\ \mathrm{V}_{\mathrm{out}}\end{bmatrix} + \begin{bmatrix}\mathrm{E}\\ \mathrm{C}\end{bmatrix}  \tag{5}\]  
In order to operate the boost converter in current control mode an appropriate controller must be designed such that it would track a reference current, by applying the definition of the sliding surface S(x,t) [19]. For n = 1 and tracking error \(\widetilde{\mathrm{x}}=(\mathrm{i}-\mathrm{i}^{\mathrm{ref}})\) i\({}^{\mathrm{ref}}\) being the desired current and i the measured current and \(\lambda\) a positive constant.