AC Filter and Grid Impedance#

This document describes continuous-time alternating current (AC) filter and grid-impedance models.

Space Vectors#

The models apply peak-valued complex space vectors, marked with boldface in the following equations. As an example, the space vector of the converter current is

(1)#\[\boldsymbol{i}^\mathrm{s}_\mathrm{c} = \frac{2}{3}\left(i_\mathrm{a} + i_\mathrm{b}\mathrm{e}^{\mathrm{j}2\pi/3} + i_\mathrm{c}\mathrm{e}^{\mathrm{j} 4\pi/3}\right)\]

where \(i_\mathrm{a}\), \(i_\mathrm{b}\), and \(i_\mathrm{c}\) are the phase currents, which may vary freely in time. In our notation, the subscript c refers to the converter-side AC quantities and the superscript s refers to the stationary coordinates. The space vector does not include the zero-sequence component, which is defined as

(2)#\[i_\mathrm{c0} = \frac{1}{3}\left(i_\mathrm{a} + i_\mathrm{b} + i_\mathrm{c}\right)\]

The space vector transformation in (1) is implemented in the function gritulator.abc2complex() and its inverse transformation in the function gritulator.complex2abc().

L Filter#

A dynamic model for an inductive L filter and inductive-resistive grid impedance is provided in the package gritulator.model._grid_filter. The model is implemented in stationary coordinates as

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

where \(\boldsymbol{i}_\mathrm{g}^\mathrm{s}\) is the grid current, \(\boldsymbol{u}_\mathrm{c}^\mathrm{s}\) is the converter voltage, \(\boldsymbol{e}_\mathrm{g}^\mathrm{s}\) is the grid voltage, \(R_\mathrm{t} = R_\mathrm{f} + R_\mathrm{g}\) is the total resistance comprising the filter series resistance \(R_\mathrm{f}\) and the grid resistance \(R_\mathrm{g}\), and \(L_\mathrm{t} = L_\mathrm{f} + L_\mathrm{g}\) is the total inductance comprising the filter inductance \(L_\mathrm{f}\) and the grid inductance \(L_\mathrm{g}\). The point of common coupling (PCC) is modeled to be between the L filter and grid impedance. The voltage at the PCC is obtained as

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

L filter and inductive-resistive grid impedance.#

LCL Filter#

A dynamic model for an inductive-capacitive-inductive (LCL) filter and inductive-resistive grid impedance is also provided in the package gritulator.model._grid_filter. The model is implemented in stationary coordinates as

(5)#\[\begin{split}\frac{\mathrm{d}\boldsymbol{i}_\mathrm{c}^\mathrm{s}}{\mathrm{d} t} = \frac{1}{L_\mathrm{fc}}(\boldsymbol{u}_\mathrm{c}^\mathrm{s} - \boldsymbol{u}_\mathrm{f}^\mathrm{s} - R_\mathrm{fc}\boldsymbol{i}_\mathrm{c}^\mathrm{s})\\ \frac{\mathrm{d}\boldsymbol{u}_\mathrm{f}^\mathrm{s}}{\mathrm{d} t} = \frac{1}{C_\mathrm{f}}(\boldsymbol{i}_\mathrm{c}^\mathrm{s} - \boldsymbol{i}_\mathrm{g}^\mathrm{s} - G_\mathrm{f}\boldsymbol{u}_\mathrm{f}^\mathrm{s})\\ \frac{\mathrm{d}\boldsymbol{i}_\mathrm{g}^\mathrm{s}}{\mathrm{d} t} = \frac{1}{L_\mathrm{t}}(\boldsymbol{u}_\mathrm{f}^\mathrm{s} - \boldsymbol{e}_\mathrm{g}^\mathrm{s} - R_\mathrm{t}\boldsymbol{i}_\mathrm{g}^\mathrm{s})\end{split}\]

where \(\boldsymbol{i}_\mathrm{c}^\mathrm{s}\) is the converter-side and \(\boldsymbol{i}_\mathrm{g}^\mathrm{s}\) is the grid-side current of the LCL filter (i.e., converter and grid current, respectively), and \(\boldsymbol{u}_\mathrm{f}^\mathrm{s}\) is the filter capacitor voltage. The converter-side and grid-side inductances of the LCL filter are \(L_\mathrm{fc}\) and \(L_\mathrm{fg}\), and their series resistances are \(R_\mathrm{fc}\) and \(R_\mathrm{fg}\), respectively. The filter capactance is \(C_\mathrm{f}\) and in parallel with it there is a conductance \(G_\mathrm{f}\). In the LCL filter model, the total grid-side indutance 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 (\(L_\mathrm{g}\), \(R_\mathrm{g}\)). The voltage at the PCC is obtained as

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

LCL filter and inductive-resistive grid impedance.#