AC Filter and Grid Impedance#

This document describes continuous-time models of an AC filter and grid impedance. The dynamics of the AC filters and grid impedance are modeled in the class motulator.grid.model.ACFilter. Based on the filter parameters the ACFilter class initializes either an L- or LCL filter. The parameters of the AC filter and grid impedance are defined in the data class motulator.grid.utils.ACFilterPars.

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.#

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.#