Induction motor model with rotor flux dynamics

Authors: Frédéric Sabot (ULB)

Reviewers: Lampros Papangelis (CRESYM)

Context

Motors are a particular kind of load that can account for a large share of the total load especially in industrialised countries. Adequate representation of motors is important, especially in short-term voltage stability studies as motors can cause fault-induced delayed voltage recovery [1].

Model use, assumptions, validity domain and limitations

The motor model can be used for short-term voltage stability studies. The assumptions made for this model are:

The rotor resistance is constant (no skin-effect or double-cage motors).
The motor is balanced.
The magnetic circuit is considered to be linear, neglecting saturation effects.
The mechanical torque varies as a constant power of the rotor speed (e.g. constant torque or quadratic torque).

The model takes into account transient and subtransient phenomena and is therefore suitable for systems with a very large motor share (>50% of the total load) or to compute the short-circuit contribution from motors.

Model description

Parameters

Parameter	Description	Unit	Typical value
$\omega_s$	Synchronous speed	$rad/s$	$314rad/s$
$R_s$	Stator resistance	$\Omega$	$0.02pu$
$L_s$	Synchronous reactance	$\Omega$	$1.8pu$
$L_p$	Transient reactance	$\Omega$	$0.12pu$
$L_{pp}$	Subtransient reactance	$\Omega$	$0.104pu$
$t_{p0}$	Transient open circuit time constant	$s$	$0.08s$
$t_{pp0}$	Subtransient open circuit time constant	$s$	$0.0021s$
$J$	Moment of inertia	$kgm^2$	0.1 to 5s
$\eta$	Exponent of the torque speed dependency	Unitless	0 to 3
$C_{l, 0}$	Initial load torque	$Nm$	N/A
$\omega_0$	Initial rotor speed	$rad/s$	N/A

Variables

Variable	Description	Unit
$V$	Stator voltage	$V$
$E_d'$	Voltage behind transient reactance d component	$V$
$E_q'$	Voltage behind transient reactance q component	$V$
$E_d''$	Voltage behind subtransient reactance d component	$V$
$E_q''$	Voltage behind subtransient reactance q component	$V$
$I_d$	Current of direct axis	$A$
$I_q$	Current of quadrature axis	$A$
$C_e$	Electrical torque	$Nm$
$C_l$	Load torque	$Nm$
SLIP	Rotor slip	Unitless
$\omega$	Rotor speed	$rad/s$

Equations

The electrical equations are described by the figure below [1].

Electrical equations of the induction motor

And is interfaced to the grid with

$V = (E_d'' + j E_q'') + (R_s + j L_{pp}) (I_d + j I_q)$

And the mechanical equations are

$2 J \frac{d\omega}{dt} = C_e - C_l$

$SLIP = \frac{\omega_s - \omega}{\omega_s}$

$C_e = E_d'' I_d + E_q'' I_q$

$C_l = C_{l, 0} \left(\frac{\omega}{\omega_0}\right)^\eta$

Open source implementations

This model has been successfully implemented in :

Software	URL	Language	Open-Source License	Last consulted date	Comments
Dynawo	Link	modelica	MPL v2.0	12/08/2024	no comment

Table of references

[1] PowerWorld. “Load Characteristic MOTORW”, https://www.powerworld.com/WebHelp/Content/TransientModels_HTML/Load%20Characteristic%20MOTORW.htm

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