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 | \(Nm\) |
| \(s\) | 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
\[ \begin{align} V = (E_d'' + j E_q'') + (R_s + j L_{pp}) (I_d + j I_q)\\ \end{align} \]
And the mechanical equations are
\[ \begin{align} 2 J \frac{d\omega}{dt} = C_e - C_l\\ SLIP = \frac{\omega_s - \omega}{\omega_s}\\ C_e = E_d'' I_d + E_q'' I_q\\ C_l = C_{l, 0} \left(\frac{\omega}{\omega_0}\right)^\eta\\ \end{align} \]
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