motulator-impedance

Mechanical impedance of machine drives and output admittance of grid converters for motulator

Documentation: https://aalto-electric-drives.github.io/motulator-impedance/

This package measures small-signal frequency responses of systems built in motulator by frequency sweeps: the system is simulated to its operating point, a small-signal excitation is injected one frequency at a time, and the response is computed with the discrete Fourier transform. The frequencies are simulated in parallel. Two responses are provided:

• Mechanical impedance Z_M(s) = −Δτ_M(s)/Δω_M(s) of a machine drive, which describes the dynamics of the machine and its control system as seen from the shaft [1]. The rotor speed is imposed, and speed excitations are superimposed on it. A drive whose mechanical impedance is passive (its phase within ±90°) is stable with any passive mechanical system.
• Output admittance of a grid converter at the point of common coupling [2], as a 2×2 matrix in synchronous coordinates. The PCC voltage is imposed by a stiff source, and voltage excitations are injected in the d- and q-axes. The admittance is the basis of impedance-based stability analysis, e.g., through its passivity index.

Installation

Install the package from PyPI (motulator is installed as a dependency):

pip install motulator-impedance

The package supports motulator 0.15, and the support is extended after each new release of motulator has been tested. For development, clone the repository and install the package with pip install -e .[dev].

Usage

The system model and the control system are built in motulator as usual. The mechanical impedance of a drive, whose rotor speed is imposed by ExternalRotorSpeed:

import motulator.drive.control.im as control
import numpy as np
from motulator.drive import model

from motulator_impedance import drive

par = model.InductionMachineInvGammaPars(
    n_p=2, R_s=60e-3, R_R=30e-3, L_sgm=2.2e-3, L_M=24.5e-3
)
mechanics = model.ExternalRotorSpeed()
mechanics.set_external_rotor_speed(40.0)  # Operating-point speed (rad/s)
mdl = model.Drive(
    model.InductionMachine(par), mechanics, model.VoltageSourceConverter(u_dc=540)
)
cfg = control.ObserverBasedVHzControllerCfg(psi_s_nom=1.04, i_s_max=170.0)
ctrl = control.VHzControlSystem(control.ObserverBasedVHzController(par, cfg))
ctrl.set_speed_ref(40.0)

meas_cfg = drive.MeasurementCfg(abs_w_e=0.3, freqs=np.geomspace(1, 500, 50))
Z_M = drive.measure_mechanical_impedance(mdl, ctrl, meas_cfg)
drive.plot_mechanical_impedance(Z_M)

The output admittance of a grid converter:

from motulator.grid import control, model

from motulator_impedance import grid

mdl = model.GridConverterSystem(
    model.VoltageSourceConverter(u_dc=650),
    model.LFilter(L_f=5e-3),
    model.ThreePhaseSource(w_g=2 * np.pi * 50, e_g=326.6),
)
cfg = control.CurrentVectorControllerCfg(i_max=30.0, L=5e-3, T_s=1 / 10e3)
ctrl = control.GridConverterControlSystem(control.CurrentVectorController(cfg))
ctrl.set_power_ref(5e3)
ctrl.set_reactive_power_ref(0.0)

meas_cfg = grid.MeasurementCfg(abs_u_e=3.0, freqs=np.geomspace(1, 10e3, 100))
Y = grid.measure_output_admittance(mdl, ctrl, meas_cfg)
grid.plot_output_admittance(Y, form="re_im")

The results are dataclasses, which can be saved, e.g., by scipy.io.savemat(path, dataclasses.asdict(Y)). Complete examples are in the example gallery.

Scope

• The frequencies are simulated in parallel processes by default (multiprocess=True), so the model and the control system must be picklable: use module-level functions or callable objects as references, not lambda functions.
• With carrier-comparison PWM, background switching harmonics can bias the measured admittance near the switching harmonics and their sidebands.
• The grid impedance and the DC bus of the system model are included in the operating point; the admittance is measured with the PCC voltage imposed.

References

1. Hartikainen, Tiitinen, Laine, Hinkkanen, "Identification of mechanical impedance of an electric machine drive for drivetrain design," Proc. IEEE IEMDC, 2023, https://doi.org/10.1109/IEMDC55163.2023.10238992
2. Määttä, Kukkola, Seppänen, Hinkkanen, "Open-source Python tool for grid converter output admittance identification," 2026, https://arxiv.org/abs/2607.10653

License

MIT