Induction motor model with rotor flux dynamics
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].
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