*2.3. LCL Filter*

The LCL filter is primarily employed in grid-interfaced solar photovoltaic systems, and its an amalgamation of *L*<sup>1</sup> + *R*1, *L*<sup>2</sup> + *R*<sup>2</sup> and C + *Rc*, which is applied in order to mitigate the harmonic effect introduced during PWM operations. Equation (7) depicts the numerical expressions:

$$\begin{cases} L\_1 \frac{di\_1}{dt} + R\_1 i\_1 = v\_i - v\_c - R\_c i\_c \\ L\_2 \frac{di\_\mathcal{S}}{dt} + R\_2 i\_\mathcal{S} = v\_c - v\_\mathcal{S} - R\_c i\_c \\ \mathcal{C} \frac{dv\_\mathcal{c}}{dt} = i\_\mathcal{c} \\ i\_1 = i\_\mathcal{S} + i\_c \end{cases} \tag{7}$$

*vg* = potential of the grid end;

*vi* = voltage of the inverter;

*R*<sup>1</sup> and *R*<sup>2</sup> represent the resistors of the inductors *L*<sup>1</sup> and *L*<sup>2</sup> consecutively.

Neglecting the resistors from the aforementioned equations, the Laplace transform of the LCL filter is derived using Equation (8).

$$G\_{\rm LCL\_{-}}\mathbf{f}(\mathbf{s}) = \frac{1}{\mathbf{s}^3 L\_1 L\_2 \mathbf{C} + \mathbf{s}(L\_1 + L\_2)}\tag{8}$$

The resonant frequency of the used filter is shown in Equation (9).

$$f\_{\text{resonant}} = \frac{1}{2\pi} \sqrt{\frac{L\_1 + L\_2}{L\_1 L\_2 C}}\tag{9}$$

The current and voltage of the grid end can be related using the transfer equation of the system model, and this is shown in Equation (10).

$$\mathbf{H(s)} = \frac{i\_g}{v\_i} = \frac{R\mathbf{C}s + 1}{L\_1 L\_2 \mathbf{C}s^3 + R\mathbf{C}(L\_1 + L\_2)s^2 + (L\_1 + L\_2)s} \tag{10}$$
