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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