AC Filter and Grid Impedance

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AC Filter and Grid Impedance#

This document describes continuous-time models of an AC filter and grid impedance.

L Filter#

Figure 1 shows a space-vector equivalent circuit of an L filter and inductive-resistive grid impedance in stationary coordinates. This model is implemented in the motulator.grid.model.LFilter class.

In general coordinates rotating at \(\omegac\), the model is

(1)#\[ \Lt\frac{\D\ic}{\D t} = \uc - \eg - \Rt\ic - \jj\omegac \Lt\ic\]

where \(\ic\) is the converter current, \(\uc\) is the converter voltage, \(\eg\) is the grid voltage, \(\Rt = \Rfc + \Rg\) is the total resistance comprising the filter series resistance \(\Rfc\) and the grid resistance \(\Rg\), and \(\Lt = \Lfc + \Lg\) is the total inductance comprising the filter inductance \(\Lfc\) and the grid inductance \(\Lg\). The point of common coupling (PCC) is modeled to be between the L filter and the grid impedance. The voltage at the PCC is

(2)#\[ \ug = \frac{\Lg(\uc - \Rfc\ic) + \Lfc(\eg + \Rg\ic)}{\Lt}\]
Diagram of L filter and grid impedance

Figure 1: L filter and inductive-resistive grid impedance in stationary coordinates.#

Diagram of L filter and grid impedance

Figure 1: L filter and inductive-resistive grid impedance in stationary coordinates.#

LCL Filter#

Figure 2 shows a space-vector equivalent circuit of an LCL filter and inductive-resistive grid impedance in stationary coordinates. This model is implemented in the motulator.grid.model.LCLFilter class.

In general coordinates rotating at \(\omegac\), the model is

(3)#\[\begin{split} \Lfc\frac{\D\ic}{\D t} &= \uc - \uf - \Rfc\ic - \jj\omegac\Lfc\ic \\ \Cf\frac{\D\uf}{\D t} &= \ic - \ig - \jj\omegac\Cf\uf \\ \Lt\frac{\D\ig}{\D t} &= \uf - \eg - \Rt\ig - \jj\omegac\Lt\ig\end{split}\]

where \(\ic\) is the converter current, \(\ig\) is the grid current, and \(\uf\) is the capacitor voltage. The converter-side and grid-side inductances of the LCL filter are \(\Lfc\) and \(\Lfg\), respectively, and their series resistances are \(\Rfc\) and \(\Rfg\), respectively. The filter capacitance is \(\Cf\). The total grid-side inductance and resistance are \(\Lt = \Lfg + \Lg\) and \(\Rt = \Rfg + \Rg\), respectively. The PCC is modeled to be between the LCL filter and the inductive-resistive grid impedance. The voltage at the PCC is

(4)#\[ \ug = \frac{\Lg(\uf - \Rfg\ig) + \Lfg(\eg + \Rg\ig)}{\Lt}\]
Diagram of LCL filter and grid impedance

Figure 2: LCL filter and inductive-resistive grid impedance in stationary coordinates.#

Diagram of LCL filter and grid impedance

Figure 2: LCL filter and inductive-resistive grid impedance in stationary coordinates.#

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