*3.1. Controller Design*

The DC-link voltage dynamics (1) can be rearranged under the following form [7,25]:

$$\frac{dV\_{dc}}{dt} = \mathbf{G}\_{dc}\mathbf{i}\_{d\mathbf{g}} - \frac{1}{\mathbf{C}\_{d\mathbf{c}}}\mathbf{i}\_{rdc} = \mathbf{G}\_{d\mathbf{c}}\mathbf{i}\_{d\mathbf{g}} + \eta\_{d\mathbf{c}}\tag{2}$$

where *ηdc* represents the uncertainties, parametric mismatch, and external disturbances of the DC-link dynamics, and it is considered to be a bounded parameter and *Gdc* = 1*Cdc* 32 *Vdg Vdc* . Let us specify the dynamics of the DC-link voltage error as [7,25]:

$$s\nu\_{\rm DC} = V\_{\rm dc} - V\_{\rm dc}^\* \tag{3}$$

The time derivative of the sliding surface *sVDC*is given as follows:

$$\frac{ds\_{V\_{dc}}}{dt} = G\_{dc}i\_{d\mathfrak{g}} + \eta\_{dc} - \frac{dV\_{dc}^\*}{dt} = \mu\_{dc} + \eta\_{dc} \tag{4}$$

where *udc* is the new control input as follows:

$$
\mu\_{dc} = G\_{dc}\dot{\imath}\_{d\text{g}} - \frac{dV\_{dc}^\*}{dt} \tag{5}
$$

While using the concept of the super twisting algorithm, a SOSM control law is designed to regulate the DC-link voltage as follows [15,25]:

$$\begin{cases} \left. u\_{dc} = -\lambda\_{dc} \right| s\_{V\_{dc}} \Big| ^{0.5} \text{sgn}(s\_{V\_{dc}}) + y\_{dc} \\ \frac{d y\_{dc}}{dt} = -a\_{dc} \text{sgn}(s\_{V\_{dc}}) \end{cases} \tag{6}$$

where *λdc* and *αdc* are parameters to be designed.

Finally, the reference direct grid current *i*∗*dg* can be derived from Equations (5) and (6) as follows:

$$\begin{cases} \begin{array}{c} i\_{d\S}^{\*} = \frac{1}{G\_{dc}} \left( -\lambda\_{dc} \left| s\_{V\_{dc}} \right|^{0.5} \text{sgn} \left( s\_{V\_{dc}} \right) + y\_{dc} + \frac{dV\_{dc}^{\*}}{dt} \right) \\ \frac{dy\_{dc}}{dt} = -\mathfrak{a}\_{dc} \text{sgn} \left( s\_{V\_{dc}} \right) \end{array} \tag{7}$$
