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STEPSS

Induction Machines

Injector models for induction machines, used as motors or generators.

The INJEC record these models are entered with, and the rules on model names, are on the Injector Models page.

INDMACH1 (inj_indmach1): Single-Cage Induction Machine

Section titled “INDMACH1 (inj_indmach1): Single-Cage Induction Machine”

Description

Section titled “Description”

A single-cage (single-rotor-circuit) induction machine model for motor loads. The machine is represented on its own MVA base (or inferred from load factor LF) with a shunt capacitor BshB_{\mathrm{sh}} to represent power factor correction. The mechanical torque is a quadratic function of rotor speed. At initialization, the model solves nonlinear algebraic equations to find the operating-point slip and flux linkages.

Single-cage induction machine block diagram. The terminal voltage drives the stator equations to give the machine currents, which with the rotor flux linkages give the electromagnetic torque. The load torque, a quadratic function of rotor speed, is subtracted and the difference is integrated through twice the inertia constant to give the rotor speed, which returns both to the load torque and to the rotor circuit as slip against the network frame. Single-cage induction machine block diagram. The terminal voltage drives the stator equations to give the machine currents, which with the rotor flux linkages give the electromagnetic torque. The load torque, a quadratic function of rotor speed, is subtracted and the difference is integrated through twice the inertia constant to give the rotor speed, which returns both to the load torque and to the rotor circuit as slip against the network frame.

Scientific Description

Section titled “Scientific Description”

The machine uses the standard dd-qq reference-frame formulation with Lss=Lsr+LlsL_{ss} = L_{sr} + L_{ls} and Lrr=Lsr+LlrL_{rr} = L_{sr} + L_{lr}. The rotor flux-linkage equations are:

dψdrdt=ω0(−RrLrrψdr+LsrRrLrriym−(ω−ωm)ψqr)\frac{d\psi_{dr}}{dt} = \omega_0\left(-\frac{R_r}{L_{rr}}\psi_{dr} + \frac{L_{sr} R_r}{L_{rr}} i_{ym} - (\omega - \omega_m)\psi_{qr}\right)

dψqrdt=ω0(−RrLrrψqr+LsrRrLrrixm+(ω−ωm)ψdr)\frac{d\psi_{qr}}{dt} = \omega_0\left(-\frac{R_r}{L_{rr}}\psi_{qr} + \frac{L_{sr} R_r}{L_{rr}} i_{xm} + (\omega - \omega_m)\psi_{dr}\right)

where ω0=2πfnom\omega_0 = 2\pi f_{\mathrm{nom}} and ωm\omega_m is the rotor mechanical speed. The equations for the stator (with shunt susceptance BshB_{\mathrm{sh}}) are:

Rsiym+(Lss−Lsr2Lrr)ω ixm+LsrLrrω ψqr=vyR_s i_{ym} + \left(L_{ss} - \frac{L_{sr}^2}{L_{rr}}\right)\omega\, i_{xm} + \frac{L_{sr}}{L_{rr}}\omega\,\psi_{qr} = v_y

Rsixm−(Lss−Lsr2Lrr)ω iym−LsrLrrω ψdr=vxR_s i_{xm} - \left(L_{ss} - \frac{L_{sr}^2}{L_{rr}}\right)\omega\, i_{ym} - \frac{L_{sr}}{L_{rr}}\omega\,\psi_{dr} = v_x

The rotor speed dynamics follow the swing equation:

dωmdt=Te−Tm2H\frac{d\omega_m}{dt} = \frac{T_e - T_m}{2H}

where the electromagnetic torque is:

Te=LsrLrr(−ψdrixm+ψqriym)T_e = \frac{L_{sr}}{L_{rr}}\left(-\psi_{dr} i_{xm} + \psi_{qr} i_{ym}\right)

and the mechanical torque is the quadratic load curve:

Tm=Tm0(A ωm2+B ωm+1−A−B)T_m = T_{m0}\left(A\,\omega_m^2 + B\,\omega_m + 1 - A - B\right)

Parameters

Section titled “Parameters”
#NameDescriptionUnit
1SNOMMachine nominal apparent power (0 = infer from LF)MVA
2RSStator resistancepu
3LlsStator leakage inductancepu
4LSRMagnetizing inductancepu
5RRRotor resistancepu
6LlrRotor leakage inductancepu
7HMachine inertia constants
8AQuadratic torque-speed coefficient
9BLinear torque-speed coefficient
10LFLoad factor (for SNOM inference)

State Variables

Section titled “State Variables”
VariableDescription
iyyy-component of stator current
ixxx-component of stator current
psidrdd-axis rotor flux linkage
psiqrqq-axis rotor flux linkage
omegamRotor mechanical speed

Observables: P, Qmot+comp, Qmot, omega, Tm

Usage Example

Section titled “Usage Example”
Show the data record 
INJEC  INDMACH1  MTR1  BUS_MV
      # FP FQ P Q: share of the power-flow injection (FP, FQ) or initial power (P in MW, Q in Mvar)
      1.  1.  0.  0.
      # SNOM: Machine nominal apparent power, 0 = infer from LF (MVA)
      10.0
      # RS: Stator resistance (pu (machine base))
      0.01
      # Lls: Stator leakage inductance (pu (machine base))
      0.10
      # LSR: Magnetizing inductance (pu (machine base))
      2.50
      # RR: Rotor resistance (pu (machine base))
      0.015
      # Llr: Rotor leakage inductance (pu (machine base))
      0.10
      # H: Inertia constant (s)
      1.5
      # A: Quadratic coefficient of load torque vs speed
      0.8
      # B: Linear coefficient of load torque vs speed
      0.1
      # LF: Load factor, initial P over SNOM, used only when SNOM = 0
      0.0
;

INDMACH2 (inj_indmach2): Double-Cage Induction Machine

Section titled “INDMACH2 (inj_indmach2): Double-Cage Induction Machine”

Description

Section titled “Description”

A double-cage (double-rotor-circuit) induction machine model following the Eurostag formulation. Two parallel rotor cages allow more accurate representation of the machine’s impedance-vs-frequency characteristic, which is especially important for the starting transient. The model structure mirrors inj_indmach1 but includes a second set of rotor flux states.

Double-cage induction machine block diagram. The structure is that of INDMACH1 with the rotor doubled: the stator equations give the machine currents, two rotor cages carry their own flux linkages, and the electromagnetic torque sums a contribution from each. The load torque is subtracted and the difference integrated through twice the inertia constant to give the rotor speed, which returns to the load torque and to both cages as slip. Double-cage induction machine block diagram. The structure is that of INDMACH1 with the rotor doubled: the stator equations give the machine currents, two rotor cages carry their own flux linkages, and the electromagnetic torque sums a contribution from each. The load torque is subtracted and the difference integrated through twice the inertia constant to give the rotor speed, which returns to the load torque and to both cages as slip.

Scientific Description

Section titled “Scientific Description”

The machine has parameters: stator resistance R1R_1, stator leakage L1L_1, magnetizing inductance LmL_m, cage-1 resistance R2R_2 and leakage L2L_2, cage-2 resistance R3R_3 and leakage L3L_3. The state vector is (ψdr1,ψqr1,ψdr2,ψqr2,ωm)(\psi_{dr1},\psi_{qr1},\psi_{dr2},\psi_{qr2},\omega_m) with flux-linkage equations for each cage:

dψdr1dt=ω0(−R2LAψdr1+LmR2LAiym−(ω−ωm)ψqr1)\frac{d\psi_{dr1}}{dt} = \omega_0\left(-\frac{R_2}{L_{A}}\psi_{dr1} + \frac{L_m R_2}{L_{A}} i_{ym} - (\omega - \omega_m)\psi_{qr1}\right)

dψdr2dt=ω0(−R3LBψdr2+LmR3LBiym−(ω−ωm)ψqr2)\frac{d\psi_{dr2}}{dt} = \omega_0\left(-\frac{R_3}{L_{B}}\psi_{dr2} + \frac{L_m R_3}{L_{B}} i_{ym} - (\omega - \omega_m)\psi_{qr2}\right)

where LA=Lm+L2L_A = L_m + L_2 and LB=Lm+L3L_B = L_m + L_3. The total electromagnetic torque combines contributions from both cages:

Te=LmLA(−ψdr1ixm+ψqr1iym)+LmLB(−ψdr2ixm+ψqr2iym)T_e = \frac{L_m}{L_A}\left(-\psi_{dr1} i_{xm} + \psi_{qr1} i_{ym}\right) + \frac{L_m}{L_B}\left(-\psi_{dr2} i_{xm} + \psi_{qr2} i_{ym}\right)

The swing equation is identical to inj_indmach1.

Parameters

Section titled “Parameters”
#NameDescriptionUnit
1SNOMMachine MVA rating (0 = infer from LF)MVA
2R1Stator resistancepu
3L1Stator leakage inductancepu
4LmMagnetizing inductancepu
5R2First cage resistancepu
6L2First cage leakage inductancepu
7R3Second cage resistancepu
8L3Second cage leakage inductancepu
9HInertia constants
10AQuadratic torque-speed coefficient
11BLinear torque-speed coefficient
12LFLoad factor (for SNOM inference)

State Variables

Section titled “State Variables”
VariableDescription
iyyy-component of stator current
ixxx-component of stator current
psidr1dd-axis flux of first rotor cage
psiqr1qq-axis flux of first rotor cage
psidr2dd-axis flux of second rotor cage
psiqr2qq-axis flux of second rotor cage
omegamRotor mechanical speed

Observables: P, Qmot+comp, Qmot, omega

Usage Example

Section titled “Usage Example”
Show the data record 
INJEC  INDMACH2  MTR2  BUS_MV
      # FP FQ P Q: share of the power-flow injection (FP, FQ) or initial power (P in MW, Q in Mvar)
      1.  1.  0.  0.
      # SNOM: Machine nominal apparent power, 0 = infer from LF (MVA)
      10.0
      # R1: Stator resistance (pu (machine base))
      0.01
      # L1: Stator leakage inductance (pu (machine base))
      0.08
      # Lm: Magnetizing inductance (pu (machine base))
      2.00
      # R2: First cage resistance (pu (machine base))
      0.02
      # L2: First cage leakage inductance (pu (machine base))
      0.06
      # R3: Second cage resistance (pu (machine base))
      0.04
      # L3: Second cage leakage inductance (pu (machine base))
      0.10
      # H: Inertia constant (s)
      1.5
      # A: Quadratic coefficient of load torque vs speed
      0.8
      # B: Linear coefficient of load torque vs speed
      0.1
      # LF: Load factor, initial P over SNOM, used only when SNOM = 0
      0.0
;

INDM1 (inj_INDM1): Alternative Induction Machine

Section titled “INDM1 (inj_INDM1): Alternative Induction Machine”

The data-file model name is INDM1 (or inj_INDM1).

Description

Section titled “Description”

An alternative single-cage induction machine model that uses the INI_indmach1 helper function for initialization. It is equivalent in physics to inj_indmach1 but implements the equations using RAMSES .txt-style model syntax with explicit state initialization calls. This can simplify parameterization when the helper function’s output is directly used.

Scientific Description

Section titled “Scientific Description”

The model equations match those of inj_indmach1, whose block diagram applies here unchanged. The key distinction is the use of the INI_indmach1 function at parameter-evaluation time to pre-compute BshB_{\mathrm{sh}}, Tm0T_{m0}, and the initial flux linkages and rotor speed, rather than solving the initialization system in Fortran. With Lss=Lsr+LlsL_{ss} = L_{sr} + L_{ls} and Lrr=Lsr+LlrL_{rr} = L_{sr} + L_{lr}, the rotor flux equations are:

dψdrdt=2πf0[−RRLrrψdr+LSRRRLrriym−(ω−ωm)ψqr]\frac{d\psi_{dr}}{dt} = 2\pi f_0 \left[ -\frac{R_R}{L_{rr}}\psi_{dr} + \frac{L_{SR} R_R}{L_{rr}} i_{ym} - (\omega - \omega_m)\psi_{qr} \right]

dψqrdt=2πf0[−RRLrrψqr+LSRRRLrrixm+(ω−ωm)ψdr]\frac{d\psi_{qr}}{dt} = 2\pi f_0 \left[ -\frac{R_R}{L_{rr}}\psi_{qr} + \frac{L_{SR} R_R}{L_{rr}} i_{xm} + (\omega - \omega_m)\psi_{dr} \right]

The rotor speed integral uses:

dωmdt=12H[LSRLrr(−ψdrixm+ψqriym)−Tm0(Aωm2+Bωm+1−A−B)]\frac{d\omega_m}{dt} = \frac{1}{2H}\left[\frac{L_{SR}}{L_{rr}}(-\psi_{dr} i_{xm} + \psi_{qr} i_{ym}) - T_{m0}(A\omega_m^2 + B\omega_m + 1 - A - B)\right]

with a lower limit of ωm≥0\omega_m \ge 0.

Parameters

Section titled “Parameters”
#NameDescriptionUnit
1SNOMMachine MVA ratingMVA
2RSStator resistancepu
3LlsStator leakage inductancepu
4LSRMagnetizing inductancepu
5RRRotor resistancepu
6LlrRotor leakage inductancepu
7HInertia constants
8AQuadratic torque-speed coefficient
9BLinear torque-speed coefficient
10LFLoad factor

Computed internally: BSH (shunt susceptance), TM0 (initial mechanical torque), LSS, LRR.

State Variables

Section titled “State Variables”
VariableDescription
Psidrdd-axis rotor flux linkage
Psiqrqq-axis rotor flux linkage
omegamRotor mechanical speed
iym, ixmStator current components (on machine base)
dPsidr, dPsiqr, domegamTime-derivative auxiliary states

Observables: omegam

Usage Example

Section titled “Usage Example”
Show the data record 
INJEC  INDM1  MTR3  BUS_MV
      # FP FQ P Q: share of the power-flow injection (FP, FQ) or initial power (P in MW, Q in Mvar)
      1.  1.  0.  0.
      # SNOM: Machine MVA rating (MVA)
      10.0
      # RS: Stator resistance (pu)
      0.01
      # Lls: Stator leakage inductance (pu)
      0.10
      # LSR: Magnetizing inductance (pu)
      2.50
      # RR: Rotor resistance (pu)
      0.015
      # Llr: Rotor leakage inductance (pu)
      0.10
      # H: Inertia constant (s)
      1.5
      # A: Quadratic torque-speed coefficient
      0.8
      # B: Linear torque-speed coefficient
      0.1
      # LF: Load factor
      0.0
;