Induction motor model with rotor flux dynamics

Author

Frédéric Sabot (ULB)

Published

December 8, 2024

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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