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Model the dynamics of three-phase asynchronous machine, also known as induction machine

The Asynchronous Machine block implements a three-phase asynchronous machine (wound rotor, single squirrel-cage, or double squirrel-cage). It 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, 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} + *d**φ*_{qs}/*dt* + *ω**φ*_{ds}

*V*_{ds} = *R*_{s}*i*_{ds} + *d**φ*_{ds}/*dt* – *ω**φ*_{qs}

*V'*_{qr} = *R'*_{r}*i'*_{qr} + *d**φ'*_{qr}/*dt* +
(*ω* – *ω*_{r})*φ'*_{dr}

*V'*_{dr} = *R'*_{r}*i'*_{dr} + *d**φ'*_{dr}/*dt* –
(*ω* – *ω*_{r})*φ'*_{qr}

*T*_{e} =
1.5*p*(*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} + *d**φ*_{qs}/*dt* + *ω**φ*_{ds}

*V*_{ds} = *R*_{s}*i*_{ds} + *d**φ*_{ds}/*dt* – *ω**φ*_{qs}

0 = *R'*_{r1}*i'*_{qr1} + *d**φ'*_{qr1}/*dt* +
(*ω* – *ω*_{r})*φ'*_{dr1}

0 = *R'*_{r1}*i'*_{dr1} + *d**φ'*_{dr1}/*dt* –
(*ω* – *ω*_{r})*φ'*_{qr1}

0 = *R'*_{r2}*i'*_{qr2} + *d**φ'*_{qr2}/*dt* +
(*ω* – *ω*_{r})*φ'*_{dr2}

0 = *R'*_{r2}*i'*_{dr2} + *d**φ'*_{dr2}/*dt* –
(*ω* – *ω*_{r})*φ'*_{qr2}

*T*_{e} =
1.5*p*(*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}

The Asynchronous Machine block parameters are defined as follows (all quantities are referred to the stator).

Parameters Common to All Models | Definition |
---|---|

R | Stator resistance and leakage inductance |

L | Magnetizing inductance |

L | Total stator inductance |

V | q axis stator voltage and current |

V | d axis stator voltage and current |

ϕ | Stator q and d axis fluxes |

ω | Angular velocity of the rotor |

Θ | Rotor angular position |

p | Number of pole pairs |

ω | Electrical angular velocity (ω |

Θ | Electrical rotor angular position (Θ |

T | Electromagnetic torque |

T | 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' | Total rotor inductance |

R' | Rotor resistance and leakage inductance |

V' | q axis rotor voltage and current |

V' | d axis rotor voltage and current |

ϕ' | Rotor q and d axis fluxes |

Parameters Specific to Double-Cage Rotor | Definition |
---|---|

R' | Rotor resistance and leakage inductance of cage 1 |

R' | Rotor resistance and leakage inductance of cage 2 |

L' | Total rotor inductances of cage 1 and 2 |

i' | q axis rotor current of cage 1 and 2 |

i' | d axis rotor current of cage 1 and 2 |

ϕ' | q and d axis rotor fluxes of cage 1 |

ϕ' | q and d axis rotor fluxes of cage 2 |

You can choose between two Asynchronous Machine blocks to specify the electrical and mechanical parameters of the model, by using the pu Units dialog box or the SI dialog box. Both blocks are modeling the same asynchronous machine model.

**Preset model**For single squirrel-cage machines, provides a set of predetermined electrical and mechanical parameters for various asynchronous machine ratings of power (HP), phase-to-phase voltage (V), frequency (Hz), and rated speed (rpm). To make this parameter available, set the

**Rotor type**parameter to`Squirrel-cage`and click**Apply**.Select one of the preset models to load the corresponding electrical and mechanical parameters in the entries of the dialog box. The preset models do not include predetermined saturation parameters.

Select

`No`if you do not want to use a preset model, or if you want to modify some of the parameters of a preset model.When you select a preset model, the electrical and mechanical parameters in the

**Parameters**tab of the dialog box become nonmodifiable (unavailable). To start from a given preset model and then modify machine parameters:Select the preset model that you want to initialize the parameters.

Change the

**Preset model**parameter value to`No`. This does not change the machine parameters. By doing so, you just break the connection with the particular preset model.Modify the machine parameters as you want, then click

**Apply**.

**Mechanical input**Select the torque applied to the shaft or the rotor speed as a Simulink

^{®}input of the block, or to represent the machine shaft by a Simscape™ rotational mechanical port.Select

**Torque Tm**to specify a torque input, in N.m or in pu, and change labeling of the block input to`Tm`. The machine speed is determined by the machine Inertia J (or inertia constant H for the pu machine) and by the difference between the applied mechanical torque Tm and the internal electromagnetic torque Te. The sign convention for the mechanical torque is: when the speed is positive, a positive torque signal indicates motor mode and a negative signal indicates generator mode.Select

**Speed w**to specify a speed input, in rad/s or in pu, and change labeling of the block input to`w`. The machine speed is imposed and the mechanical part of the model (Inertia J) is ignored. Using the speed as the mechanical input allows modeling a mechanical coupling between two machines.The next figure indicates how to model a stiff shaft interconnection in a motor-generator set when friction torque is ignored in machine 2. The speed output of machine 1 (motor) is connected to the speed input of machine 2 (generator), while machine 2 electromagnetic torque output Te is applied to the mechanical torque input Tm of machine 1. The Kw factor takes into account speed units of both machines (pu or rad/s) and gear box ratio w2/w1. The KT factor takes into account torque units of both machines (pu or N.m) and machine ratings. Also, as the inertia J2 is ignored in machine 2, J2 referred to machine 1 speed must be added to machine 1 inertia J1.

Select

**Mechanical rotational port**to add to the block a Simscape mechanical rotational port that allows connection of the machine shaft with other Simscape blocks having mechanical rotational ports. The Simulink input representing the mechanical torque Tm or the speed`w`of the machine is then removed from the block.The next figure indicates how to connect an Ideal Torque Source block from the Simscape library to the machine shaft to represent the machine in motor mode, or in generator mode, when the rotor speed is positive.

**Rotor type**Specifies the type of rotor:

`Wound`,`Squirrel-cage`, or`Double squirrel-cage`.**Reference frame**Specifies the reference frame that is used to convert input voltages (abc reference frame) to the dq reference frame, and output currents (dq reference frame) to the abc reference frame. You can choose among the following reference frame transformations:

`Rotor`(Park transformation)`Stationary`(Clarke or αβ transformation)`Synchronous`

The following relationships describe the abc-to-dq reference frame transformations applied to the Asynchronous Machine phase-to-phase voltages.

In the preceding equations, Θ is the angular position of the reference frame, while

*β*=*θ*–*θ*is the difference between the position of the reference frame and the position (electrical) of the rotor. Because the machine windings are connected in a three-wire Y configuration, there is no homopolar (0) component. This configuration also justifies that two line-to-line input voltages are used inside the model instead of three line-to-neutral voltages. The following relationships describe the dq-to-abc reference frame transformations applied to the Asynchronous Machine phase currents._{r}The following table shows the values taken by Θ and β in each reference frame (Θ

_{e}is the position of the synchronously rotating reference frame).Reference Frame

Θ

β

Rotor

Θ

_{r}0

Stationary

0

−Θ

_{r}Synchronous

Θ

_{e}Θ

_{e}− Θ_{r}The choice of reference frame affects the waveforms of all dq variables. It also affects the simulation speed and in certain cases the accuracy of the results. The following guidelines are suggested in [1]:

Use the stationary reference frame if the stator voltages are either unbalanced or discontinuous and the rotor voltages are balanced (or 0).

Use the rotor reference frame if the rotor voltages are either unbalanced or discontinuous and the stator voltages are balanced.

Use either the stationary or synchronous reference frames if all voltages are balanced and continuous.

**Use signal names to identify bus labels**When this check box is selected, the measurement output uses the signal names to identify the bus labels. Select this option for applications that require bus signal labels to have only alphanumeric characters.

When this check box is cleared, the measurement output uses the signal definition to identify the bus labels. The labels contain nonalphanumeric characters that are incompatible with some Simulink applications.

This tab contains the electrical parameters of the machine.
To estimate the electrical parameters of a double-cage asynchronous
machine based on standard manufacturer specifications, you can use
the `power_AsynchronousMachineParams` function.

**Nominal power, voltage (line-line), and frequency**The nominal apparent power Pn (VA), RMS line-to-line voltage Vn (V), and frequency fn (Hz).

**Stator resistance and inductance**The stator resistance Rs (Ω or pu) and leakage inductance Lls (H or pu).

**Rotor resistance and inductance**The rotor resistance Rr' (Ω or pu) and leakage inductance Llr' (H or pu), both referred to the stator. This parameter is visible only when the

**Rotor type**parameter on the**Configuration**tab is set to`Wound`or`Squirrel-cage`.**Cage 1 resistance and inductance**The rotor resistance Rr1' (Ω or pu) and leakage inductance Llr1' (H or pu), both referred to the stator. This parameter is visible only when the

**Rotor type**parameter on the**Configuration**tab is set to`Double squirrel-cage`.**Cage 2 resistance and inductance**The rotor resistance Rr2' (Ω or pu) and leakage inductance Llr2' (H or pu), both referred to the stator. This parameter is visible only when the

**Rotor type**parameter on the**Configuration**tab is set to`Double squirrel-cage`.**Mutual inductance**The magnetizing inductance Lm (H or pu).

**Inertia constant, friction factor, and pole pairs**For the

**SI units**dialog box: the combined machine and load inertia coefficient J (kg.m^{2}), combined viscous friction coefficient F (N.m.s), and pole pairs p. The friction torque Tf is proportional to the rotor speed ω (Tf = F.w).For the

**pu units**dialog box: the inertia constant H (s), combined viscous friction coefficient F (pu), and pole pairs p.**Initial conditions**Specifies the initial slip s, electrical angle Θe (degrees), stator current magnitude (A or pu), and phase angles (degrees):

[slip, th, i

_{as}, i_{bs}, i_{cs}, phase_{as}, phase_{bs}, phase_{cs}]If the

**Rotor type**parameter is set to`Wound`, you can also specify optional initial values for the rotor current magnitude (A or pu), and phase angles (degrees):[slip, th, i

_{as}, i_{bs}, i_{cs}, phase_{as}, phase_{bs}, phase_{cs}, i_{ar}, i_{br}, i_{cr}, phase_{ar}, phase_{br}, phase_{cr}]When the

**Rotor type**parameter is set to`Squirrel-cage`, the initial conditions can be computed by the Load Flow tool or the Machine Initialization tool in the Powergui block.**Simulate saturation**Specifies whether magnetic saturation of the rotor and stator iron is simulated or not.

**[i;v] (pu)**Specifies the no-load saturation curve parameters. Magnetic saturation of the stator and rotor iron (saturation of the mutual flux) is modeled by a piecewise linear relationship specifying points of the no-load saturation curve. The first row of this matrix contains the values of stator currents. The second row contains values of corresponding terminal voltages (stator voltages). The first point (first column of the matrix) must be different from [0,0]. This point corresponds to the point where the effect of saturation begins.

You must select the

**Simulate saturation**check box to simulate saturation. If you do not select the**Simulate saturation**check box, the relationship between the stator current and the stator voltage is linear.Click

**Plot**to view the specified no-load saturation curve.

**Sample time (−1 for inherited)**Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to

`-1`.**Discrete solver model**Specifies the integration method used by the block when the

**Solver type**parameter of the Powergui block is set to`Discrete`. The choices are:`Forward Euler`(default),`Trapezoidal non iterative`, and`Trapezoidal iterative (alg. loop)`.For more information on what method you should use in your application, see Simulating Discretized Electrical Systems.

The parameters on this tab are used by the Load Flow tool of the Powergui block. These load flow parameters are used for model initialization only. They have no impact on the block model or on the simulation performance.

The stator terminals of the Asynchronous Machine block 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.

`Tm`The Simulink input of the block is the mechanical torque at the machine's shaft. When the input is a positive Simulink signal, the asynchronous machine behaves as a motor. When the input is a negative signal, the asynchronous machine behaves as a generator.

When you use the SI parameters mask, the input is a signal in N.m, otherwise it is in pu.

`w`The alternative block input (depending on the value of the

**Mechanical input**parameter) is the machine speed. When you use the SI parameters mask, the input is a signal in rad/s or in pu.`m`The Simulink output of the block is a vector containing measurement signals. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library. Depending on the type of mask that you use, the units are in SI or in pu. The cage 2 rotor signals return null signal when the

**Rotor type**parameter on the**Configuration**tab is set to`Wound`or`Squirrel-cage`.Name

Definition

Units

iar

Rotor current ir_a

A or pu

ibr

Rotor current ir_b

A or pu

icr

Rotor current ir_c

A or pu

iqr

Rotor current iq

A or pu

idr

Rotor current id

A or pu

phiqr

Rotor flux phir_q

V.s or pu

phidr

Rotor flux phir_d

V.s or pu

vqr

Rotor voltage Vr_q

V or pu

vdr

Rotor voltage Vr_d

V or pu

iar2

Cage 2 rotor current ir_a

A or pu

ibr2

Cage 2 rotor current ir_b

A or pu

icr2

Cage 2 rotor current ir_c

A or pu

iqr2

Cage 2 rotor current iq

A or pu

idr2

Cage 2 rotor current id

A or pu

phiqr2

Cage 2 rotor flux phir_q

V.s or pu

phidr2

Cage 2 rotor flux phir_d

V.s or pu

ias

Stator current is_a

A or pu

ibs

Stator current is_b

A or pu

ics

Stator current is_c

A or pu

iqs

Stator current is_q

A or pu

ids

Stator current is_d

A or pu

phiqs

Stator flux phis_q

V.s or pu

phids

Stator flux phis_d

V.s or pu

vqs

Stator voltage vs_q

V or pu

vds

Stator voltage vs_d

V or pu

w

Rotor speed

rad/s

Te

Electromagnetic torque Te

N.m or pu

theta

Rotor angle thetam

rad

The Asynchronous Machine block does not include a representation of the saturation of leakage fluxes. You must be careful when you connect ideal sources to the machine's stator. 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 use a small parasitic resistive load, connected at the machine terminals, to avoid numerical oscillations. Large sample times require larger loads. The optimum resistive load is proportional to the sample time. Remember that 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 `power_pwm``power_pwm` example
illustrates the use of the Asynchronous Machine block in motor mode.
It 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 second.

The parameters of the machine are those found in the preceding **SI Units** dialog box above, except for the stator
leakage inductance, which is set to twice its 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_pwmpower_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_satpower_asm_sat` example
illustrates the effect of saturation of the Asynchronous Machine block.

Two identical three-phase motors (50 HP, 460 V, 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.

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