Model dynamics of three-phase asynchronous machine, also known as induction machine, in SI or pu units
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The Asynchronous Machine SI Units and Asynchronous Machine pu Units blocks implement a three-phase asynchronous machine (wound rotor, squirrel cage, or double squirrel cage) modeled in a selectable dq reference frame (rotor, stator, or synchronous). Stator and rotor windings are connected in wye to an internal neutral point. The block operates in either generator or motor mode. The mode of operation is dictated by the sign of the mechanical torque:
If Tm is positive, the machine acts as a motor.
If Tm is negative, the machine acts as a generator.
The electrical part of the machine is represented by a fourth-order (or sixth-order for the double squirrel-cage machine) state-space model, and the mechanical part by a second-order system. All electrical variables and parameters are referred to the stator, which is indicated by the prime signs in the following machine equations. All stator and rotor quantities are in the arbitrary two-axis reference frame (dq frame). The subscripts used are defined in this table.
Subscript | Definition |
---|---|
d | d-axis quantity |
q | q-axis quantity |
r | Rotor quantity (wound-rotor or single-cage) |
r1 | Cage 1 rotor quantity (double-cage) |
r2 | Cage 2 rotor quantity (double-cage) |
s | Stator quantity |
l | Leakage inductance |
m | Magnetizing inductance |
V_{qs} = R_{s}i_{qs} + dφ_{qs}/dt + ωφ_{ds}
V_{ds} = R_{s}i_{ds} + dφ_{ds}/dt – ωφ_{qs}
V'_{qr} = R'_{r}i'_{qr} + dφ'_{qr}/dt + (ω – ω_{r})φ'_{dr}
V'_{dr} = R'_{r}i'_{dr} + dφ'_{dr}/dt – (ω – ω_{r})φ'_{qr}
T_{e} = 1.5p(φ_{ds}i_{qs} – φ_{qs}i_{ds})
ω — Reference frame angular velocity
ω_{r} — Electrical angular velocity
φ_{qs} = L_{s}i_{qs} + L_{m}i'_{qr}
φ_{ds} = L_{s}i_{ds} + L_{m}i'_{dr}
φ'_{qr} = L'_{r}i'_{qr} + L_{m}i_{qs}
φ'_{dr} = L'_{r}i'_{dr} + L_{m}i_{ds}
L_{s} = L_{ls} + L_{m}
L'_{r} = L'_{lr} + L_{m}
V_{qs} = R_{s}i_{qs} + dφ_{qs}/dt + ωφ_{ds}
V_{ds} = R_{s}i_{ds} + dφ_{ds}/dt – ωφ_{qs}
0 = R'_{r1}i'_{qr1} + dφ'_{qr1}/dt + (ω – ω_{r})φ'_{dr1}
0 = R'_{r1}i'_{dr1} + dφ'_{dr1}/dt – (ω – ω_{r})φ'_{qr1}
0 = R'_{r2}i'_{qr2} + dφ'_{qr2}/dt + (ω – ω_{r})φ'_{dr2}
0 = R'_{r2}i'_{dr2} + dφ'_{dr2}/dt – (ω – ω_{r})φ'_{qr2}
T_{e} = 1.5p(φ_{ds}i_{qs} – φ_{qs}i_{ds})
φ_{qs} = L_{s}i_{qs} + L_{m}(i'_{qr1} + i'_{qr2})
φ_{ds} = L_{s}i_{ds} + L_{m}(i'_{dr1} + i'_{dr2})
φ'_{qr1} = L'_{r1}i'_{qr1} + L_{m}i_{qs}
φ'_{dr1} = L'_{r1}i'_{dr1} + L_{m}i_{ds}
φ'_{qr2} = L'_{r2}i'_{qr2} + L_{m}i_{qs}
φ'_{dr2} = L'_{r2}i'_{dr2} + L_{m}i_{ds}
L_{s} = L_{ls} + L_{m}
L'_{r1} = L'_{lr1} + L_{m}
L'_{r2} = L'_{lr2} + L_{m}
$$\begin{array}{c}\frac{d}{dt}{\omega}_{m}=\frac{1}{2H}\left({T}_{e}-F{\omega}_{m}-{T}_{m}\right)\\ \frac{d}{dt}{\theta}_{m}={\omega}_{m}\end{array}$$
The Asynchronous Machine block parameters are defined in the table. All quantities are referred to the stator.
Parameters Common to All Models | Definition |
---|---|
R_{s}, L_{ls} | Stator resistance and leakage inductance |
L_{m} | Magnetizing inductance |
L_{s} | Total stator inductance |
V_{qs}, i_{qs} | q-axis stator voltage and current |
V_{ds}, i_{ds} | d-axis stator voltage and current |
ϕ_{qs}, ϕϕ_{ds} | Stator q-axis and d-axis fluxes |
ω_{m} | Angular velocity of the rotor |
Θ_{m} | Rotor angular position |
p | Number of pole pairs |
ω_{r} | Electrical angular velocity (ω_{m} × p) |
Θ_{r} | Electrical rotor angular position (Θ_{m} × p) |
T_{e} | Electromagnetic torque |
T_{m} | Shaft mechanical torque |
J | Combined rotor and load inertia coefficient. Set to infinite to simulate locked rotor. |
H | Combined rotor and load inertia constant. Set to infinite to simulate locked rotor. |
F | Combined rotor and load viscous friction coefficient |
Parameters Specific to Single-Cage or Wound Rotor | Definition |
---|---|
L'_{r} | Total rotor inductance |
R'_{r}, L'_{lr} | Rotor resistance and leakage inductance |
V'_{qr}, i'_{qr} | q-axis rotor voltage and current |
V'_{dr}, i'_{dr} | d-axis rotor voltage and current |
ϕ'_{qr}, ϕ'_{dr} | Rotor q-axis and d axis fluxes |
Parameters Specific to Double-Cage Rotor | Definition |
---|---|
R'_{r1}, L'_{lr1} | Rotor resistance and leakage inductance of cage 1 |
R'_{r2}, L'_{lr2} | Rotor resistance and leakage inductance of cage 2 |
L'_{r1}, L'_{r2} | Total rotor inductances of cage 1 and 2 |
i'_{qr1}, i'_{qr2} | q-axis rotor current of cage 1 and 2 |
i'_{dr1}, i'_{dr2} | d-axis rotor current of cage 1 and 2 |
ϕ'_{qr1}, ϕ'_{dr1} | q-axis and d-axis rotor fluxes of cage 1 |
ϕ'_{qr2}, ϕ'_{dr2} | q-axis and d-axis rotor fluxes of cage 2 |
The Asynchronous Machine blocks do not include a representation of the saturation of leakage fluxes. Be careful when you connect ideal sources to the stator of the machine. If you choose to supply the stator via a three-phase, Y-connected infinite voltage source, you must use three sources connected in Y. However, if you choose to simulate a delta source connection, you must use only two sources connected in series.
When you use Asynchronous Machine blocks in discrete systems, you might have to connect a small parasitic resistive load at the machine terminals to avoid numerical oscillations. Large sample times require larger loads. The optimum resistive load is proportional to the sample time. With a 25 μs time step on a 60 Hz system, the minimum load is approximately 2.5% of the machine nominal power. For example, a 200 MVA asynchronous machine in a power system discretized with a 50 μs sample time requires approximately 5% of resistive load or 10 MW. If the sample time is reduced to 20 μs, a resistive load of 4 MW is sufficient.
The stator terminals of the Asynchronous Machine blocks are identified by the letters A, B, and C. The rotor terminals are identified by the letters a, b, and c. The neutral connections of the stator and rotor windings are not available. Three-wire Y connections are assumed.
The power_pwm
example uses a Asynchronous
Machine block in motor mode. The example consists of an asynchronous
machine in an open-loop speed control system.
The machine rotor is short-circuited, and the stator is fed by a PWM inverter built with Simulink blocks and interfaced to the Asynchronous Machine block through the Controlled Voltage Source block. The inverter uses sinusoidal pulse-width modulation. The base frequency of the sinusoidal reference wave is set at 60 Hz and the triangular carrier wave frequency is set at 1980 Hz. This frequency corresponds to a frequency modulation factor m_{f} of 33 (60 Hz x 33 = 1980).
The 3 HP machine is connected to a constant load of nominal value (11.9 N.m). It is started and reaches the set point speed of 1.0 pu at t = 0.9 seconds.
The parameters of the machine are the same as the Asynchronous Machine SI Units block, except for the stator leakage inductance, which is set to twice the normal value to simulate a smoothing inductor placed between the inverter and the machine. Also, the stationary reference frame was used to obtain the results shown.
Open the power_pwm
example. In the simulation parameters, a small relative tolerance is required
because of the high switching rate of the inverter.
Run the simulation and observe the machine's speed and torque.
The first graph shows the machine's speed going from 0 to 1725 rpm (1.0 pu). The second graph shows the electromagnetic torque developed by the machine. Because the stator is fed by a PWM inverter, a noisy torque is observed.
However, this noise is not visible in the speed because it is filtered out by the machine's inertia, but it can be seen in the stator and rotor currents.
Look at the output of the PWM inverter. Because nothing of interest can be seen at the simulation time scale, the graph concentrates on the last moments of the simulation.
The power_asm_sat
example illustrates the effect of saturation of
the Asynchronous Machine block.
Two identical three-phase motors (50 HP, 460 V, and 1800 rpm) are simulated, with and without saturation, to observe the saturation effects on the stator currents. Two different simulations are realized in the example.
The first simulation is the no-load steady-state test. This table contains the values of the saturation parameters and the measurements obtained by simulating different operating points on the saturated motor (no-load and in steady-state).
Saturation Parameters | Measurements | ||
---|---|---|---|
Vsat (Vrms L-L) | Isat (peak A) | Vrms L-L | Is_A (peak A) |
- | - | 120 | 7.322 |
230 | 14.04 | 230 | 14.03 |
- | - | 250 | 16.86 |
- | - | 300 | 24.04 |
322 | 27.81 | 322 | 28.39 |
- | - | 351 | 35.22 |
- | - | 382 | 43.83 |
414 | 53.79 | 414 | 54.21 |
- | - | 426 | 58.58 |
- | - | 449 | 67.94 |
460 | 72.69 | 460 | 73.01 |
- | - | 472 | 79.12 |
- | - | 488 | 88.43 |
506 | 97.98 | 506 | 100.9 |
- | - | 519 | 111.6 |
- | - | 535 | 126.9 |
- | - | 546 | 139.1 |
552 | 148.68 | 552 | 146.3 |
- | - | 569 | 169.1 |
- | - | 581 | 187.4 |
598 | 215.74 | 598 | 216.5 |
- | - | 620 | 259.6 |
- | - | 633 | 287.8 |
644 | 302.98 | 644 | 313.2 |
- | - | 659 | 350 |
- | - | 672 | 383.7 |
- | - | 681 | 407.9 |
690 | 428.78 | 690 | 432.9 |
The next graph illustrates these results and shows the accuracy of the saturation model. The measured operating points fit well the curve that is plotted from the saturation parameters data.
You can observe the other effects of saturation on the stator currents by running the simulation with a blocked rotor or with many different values of load torque.
[1] Krause, P.C., O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery, IEEE^{®} Press, 2002.
[2] Mohan, N., T.M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications, and Design, John Wiley & Sons, Inc., New York, 1995, Section 8.4.1.