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#

A dynamic model of an L filter and inductive-resistive grid impedance is implemented in the class motulator.grid.model.LFilter. The model in stationary coordinates is

(1)#\[L_\mathrm{t}\frac{\mathrm{d}\boldsymbol{i}_\mathrm{c}^\mathrm{s}}{\mathrm{d} t} = \boldsymbol{u}_\mathrm{c}^\mathrm{s} - \boldsymbol{e}_\mathrm{g}^\mathrm{s} - R_\mathrm{t}\boldsymbol{i}_\mathrm{c}^\mathrm{s}\]

where \(\boldsymbol{i}_\mathrm{c}^\mathrm{s}\) is the converter current, \(\boldsymbol{u}_\mathrm{c}^\mathrm{s}\) is the converter voltage, \(\boldsymbol{e}_\mathrm{g}^\mathrm{s}\) is the grid voltage, \(R_\mathrm{t} = R_\mathrm{fc} + R_\mathrm{g}\) is the total resistance comprising the filter series resistance \(R_\mathrm{fc}\) and the grid resistance \(R_\mathrm{g}\), and \(L_\mathrm{t} = L_\mathrm{fc} + L_\mathrm{g}\) is the total inductance comprising the filter inductance \(L_\mathrm{fc}\) and the grid inductance \(L_\mathrm{g}\). 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)#\[\boldsymbol{u}_\mathrm{g}^\mathrm{s} = \frac{L_\mathrm{g}(\boldsymbol{u}_\mathrm{c}^\mathrm{s} - R_\mathrm{fc}\boldsymbol{i}_\mathrm{c}^\mathrm{s}) + L_\mathrm{fc}(\boldsymbol{e}_\mathrm{g}^\mathrm{s} + R_\mathrm{g}\boldsymbol{i}_\mathrm{c}^\mathrm{s})}{L_\mathrm{t}}\]
Diagram of L filter and grid impedance

L filter and inductive-resistive grid impedance.#

Diagram of L filter and grid impedance

L filter and inductive-resistive grid impedance.#

LCL Filter#

A dynamic model of an LCL filter and inductive-resistive grid impedance is implemented in the class motulator.grid.model.LCLFilter. The model in stationary coordinates is

(3)#\[\begin{split}L_\mathrm{fc}\frac{\mathrm{d}\boldsymbol{i}_\mathrm{c}^\mathrm{s}}{\mathrm{d} t} &= \boldsymbol{u}_\mathrm{c}^\mathrm{s} - \boldsymbol{u}_\mathrm{f}^\mathrm{s} - R_\mathrm{fc}\boldsymbol{i}_\mathrm{c}^\mathrm{s}\\ C_\mathrm{f}\frac{\mathrm{d}\boldsymbol{u}_\mathrm{f}^\mathrm{s}}{\mathrm{d} t} &= \boldsymbol{i}_\mathrm{c}^\mathrm{s} - \boldsymbol{i}_\mathrm{g}^\mathrm{s}\\ L_\mathrm{t}\frac{\mathrm{d}\boldsymbol{i}_\mathrm{g}^\mathrm{s}}{\mathrm{d} t} &= \boldsymbol{u}_\mathrm{f}^\mathrm{s} - \boldsymbol{e}_\mathrm{g}^\mathrm{s} - R_\mathrm{t}\boldsymbol{i}_\mathrm{g}^\mathrm{s}\end{split}\]

where \(\boldsymbol{i}_\mathrm{c}^\mathrm{s}\) is the converter current, \(\boldsymbol{i}_\mathrm{g}^\mathrm{s}\) is the grid current, and \(\boldsymbol{u}_\mathrm{f}^\mathrm{s}\) is the capacitor voltage. The converter-side and grid-side inductances of the LCL filter are \(L_\mathrm{fc}\) and \(L_\mathrm{fg}\), respectively, and their series resistances are \(R_\mathrm{fc}\) and \(R_\mathrm{fg}\), respectively. The filter capacitance is \(C_\mathrm{f}\). The total grid-side inductance and resistance are \(L_\mathrm{t} = L_\mathrm{fg} + L_\mathrm{g}\) and \(R_\mathrm{t} = R_\mathrm{fg} + R_\mathrm{g}\), respectively. The PCC is modeled to be between the LCL filter and the inductive-resistive grid impedance. The voltage at the PCC is

(4)#\[\boldsymbol{u}_\mathrm{g}^\mathrm{s} = \frac{L_\mathrm{g}(\boldsymbol{u}_\mathrm{f}^\mathrm{s} - R_\mathrm{fg}\boldsymbol{i}_\mathrm{g}^\mathrm{s}) + L_\mathrm{fg}(\boldsymbol{e}_\mathrm{g}^\mathrm{s} + R_\mathrm{g}\boldsymbol{i}_\mathrm{g}^\mathrm{s})}{L_\mathrm{t}}\]
Diagram of LCL filter and grid impedance

LCL filter and inductive-resistive grid impedance.#

Diagram of LCL filter and grid impedance

LCL filter and inductive-resistive grid impedance.#

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