Induction machine
Induction machine model for detailed representation of electrical loads.
Disclaimer: This model has only been tested for the power flow problem with ACP formulation.
Parameters
Set of parameters used to model the induction machine as defined in the input data
| name | symb. | unit | type | default | definition |
|---|---|---|---|---|---|
| index | $im$ | - | Int | - | unique index of the induction machine |
| im_bus | $i$ | - | Int | - | unique index of the bus to which the induction machine is connected to |
| P_ag | $P_{ag}$ | p.u. | Real | - | Starting value for active power of induction machine - positive for consumption |
| Q_ag | $Q_{ag}$ | p.u. | Real | - | Starting value for active power of induction machine - positive for consumption |
| Pacmin | $\underline{P_{im}}$ | p.u. | Real | - | minimum stable operating power of the induction machine |
| Pacmax | $\overline{P_{im}}$ | p.u. | Real | - | maximum power rating of the induction machine |
| Pacrated | $P_{im}$ | p.u. | Real | - | minimum reactive power of the induction machine |
| status | $\delta_{im}$ | - | Int | - | Status indicator of the induction machine |
| x_m | $x_{m}$ | p.u. | Real | - | Magnetizing inductance of induction machine |
| x_sl | $x_{sl}$ | p.u. | Real | - | Stator leakage inductance of induction machine |
| x_rl | $x_{rl}$ | p.u. | Real | - | Rotor leakage inductance of induction machine |
| r_s | $r_{s}$ | p.u. | Real | - | Stator resistance of induction machine |
| r_r | $r_{r}$ | p.u. | Real | - | Rotor resistance of induction machine |
| torque | - | - | - | - | Torque model parameters of induction machine |
Torque parameter
The torque model is implemented as $T(\omega) = T_0 * (A*\omega^m+B \omega + C)$. This representation includes both quadratic models (m=2) or power functions (B=C=0) (see Kundur).
| Name | Symbol | Unit | Type | Default | Description |
|---|---|---|---|---|---|
| T_0 | $T_0$ | p.u. | Real | - | Per-unit torque scaling factor (base torque is approximately equal to the system base power) |
| A | $A$ | $\mathrm{s}^2/\mathrm{rad}^2$ | Real | - | Quadratic coefficient of the mechanical torque characteristic (load component typical for centrifugal devices or aerodynamic drag) |
| B | $B$ | $\mathrm{s}/\mathrm{rad}$ | Real | - | Linear coefficient of the mechanical torque characteristic (friction load component) |
| C | $C$ | - | Real | - | Constant coefficient of the mechanical torque characteristic (static load component) |
| m | $m$ | - | Real | - | Mechanical torque exponent describing the load type (see Kundur) |
Variables
The main optimisation variables of interest are:
| name | symb. | unit | formulation | definition |
|---|---|---|---|---|
| pg | $P_{g}$ | p.u. | ACP | Active power of induction machine g |
| qg | $Q_{g}$ | p.u. | ACP | Reactive power of induction machine g |
Constraints
See Kundur or Van Cutsem for detailed derivation of equations.
Stator constrains
PowerModelsACDC.constraint_im_stator — Function
Im stator constraints (Based on transformer formulation)
\[p_{im,s,fr} = g v_{m,fr}^{2} - g v_{m,fr} v_{m,to}\cos\left(v_{a,fr}-v_{a,to}\right) - b v_{m,fr} v_{m,to}\sin\left(v_{a,fr}-v_{a,to}\right)\]
\[q_{im,s,fr} = - b v_{m,fr}^{2} + b v_{m,fr} v_{m,to}\cos\left(v_{a,fr}-v_{a,to}\right) - g v_{m,fr} v_{m,to}\sin\left(v_{a,fr}-v_{a,to}\right)\]
\[p_{im,s,to} = g v_{m,to}^{2} - g v_{m,to} v_{m,fr}\cos\left(v_{a,to}-v_{a,fr}\right) - b v_{m,to} v_{m,fr}\sin\left(v_{a,to}-v_{a,fr}\right)\]
\[q_{im,s,to} = - b v_{m,to}^{2} + b v_{m,to} v_{m,fr}\cos\left(v_{a,to}-v_{a,fr}\right) - g v_{m,to} v_{m,fr}\sin\left(v_{a,to}-v_{a,fr}\right)\]
Rotor inductance constraints
PowerModelsACDC.constraint_im_rotor_inductance — Function
IM rotor inductance constraints
\[-p_{im,ag} = g_c v_{m,ag}^{2} - g_c v_{m,ag} v_{m,m}\cos\left(v_{a,ag}-v_{a,m}\right) - b_c v_{m,ag} v_{m,m}\sin\left(v_{a,ag}-v_{a,m}\right)\]
\[-q_{im,ag} = - b_c v_{m,ag}^{2} + b_c v_{m,ag} v_{m,m}\cos\left(v_{a,ag}-v_{a,m}\right) - g_c v_{m,ag} v_{m,m}\sin\left(v_{a,ag}-v_{a,m}\right)\]
\[p_{im,ri,f} = g_c v_{m,m}^{2} - g_c v_{m,m} v_{m,ag}\cos\left(v_{a,m}-v_{a,ag}\right) - b_c v_{m,m} v_{m,ag}\sin\left(v_{a,m}-v_{a,ag}\right)\]
\[q_{im,ri,f} = - b_c v_{m,m}^{2} + b_c v_{m,m} v_{m,ag}\cos\left(v_{a,m}-v_{a,ag}\right) - g_c v_{m,m} v_{m,ag}\sin\left(v_{a,m}-v_{a,ag}\right)\]
Magnetistation branch constraints
PowerModelsACDC.constraint_im_magnetisation — Function
IM magnetisation constraint
\[p_{im,ri,f}+p_{im,s,to}=0\]
\[q_{im,ri,f}+q_{im,s,to}+b_v v_{m,m}^{2}=0\]
Slip constraints
PowerModelsACDC.constraint_im_slip — Function
IM slip constraints (slip from balance air-gap power with mechanical torque, see Van Cutsem Voltage stability)
\[T_0\left(A(1-s)^m+B(1-s)+C\right) = \frac{v_{m,ag}^{2}s}{r_r}\]
\[p_{im,ag} = \frac{v_{m,ag}^{2}s}{r_r}\]
\[q_{im,ag}=0\]