# Synchronous Machine Model 2.1

Synchronous machine with simplified transformation, simplified representation, and fundamental or standard parameterization

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• Simscape / Electrical / Electromechanical / Synchronous

## Description

The Synchronous Machine Model 2.1 block models a synchronous machine with one field winding and one damper on the d-axis and one damper on the q-axis. You use fundamental or standard parameters to define the characteristics of the machine. This block contains a dq Park transformation, so use it only for balanced operation.

### Synchronous Machine Initialization Using Load-Flow Target Values

If the block is in a network that is compatible with the frequency-time simulation mode, you can perform a load-flow analysis on the network. A load-flow analysis provides steady-state values that you can use to initialize the machine.

For more information, see Perform a Load-Flow Analysis Using Simscape Electrical and Frequency and Time Simulation Mode. For an example that shows how initialize an synchronous machine using data from a load flow analysis, see Synchronous Machine Initialization with Loadflow.

### Equations

The synchronous machine equations are expressed with respect to a rotating reference frame, defined by

`${\theta }_{e}\left(t\right)=N{\theta }_{r}\left(t\right),$`

where:

• θe is the electrical angle.

• N is the number of pole pairs.

• θr is the rotor angle.

The Park transformation maps the synchronous machine equations to the rotating reference frame with respect to the electrical angle. The Park transformation is defined by

`${P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}+\frac{2\pi }{3}\right)\end{array}\right].$`

The Park transformation is used to define the per-unit synchronous machine equations. The stator voltage equations are defined by

`${e}_{d}={e}_{d}^{"}-{R}_{a}{i}_{d}+{x}_{q}^{"}{i}_{q}$`

and

`${e}_{q}={e}_{q}^{"}-{x}_{d}^{"}{i}_{d}-{R}_{a}{i}_{q},$`

where:

• e”d and e”q are the d-axis and q-axis voltages behind subtransient reactances.

• Ra is the stator resistance.

• id and iq are the d-axis and q-axis stator currents, defined by

`$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$`

ia, ib, and ic are the stator currents flowing from port ~ to port n.

• x”d and x”q are the d-axis and q-axis subtransient reactances.

• ed and eq are the d-axis and q-axis stator voltages, defined by

`$\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$`

va, vb, and vc are the stator voltages measured from port ~ to neutral port n.

The rotor voltage equation is defined by

`${e}_{fd}={R}_{fd}\cdot {i}_{fd},$`

where:

• Rfd is the resistance of rotor field circuit.

• ifd is the per-unit field current using the synchronous machine model reciprocal per-unit system.

• efd is the per-unit field voltage using the synchronous machine model reciprocal per-unit system.

The voltage-behind-transient-reactance equations are defined by

`$\frac{d{e}_{d}^{"}}{dt}=\frac{\left({x}_{q}-{x}_{q}^{"}\right){i}_{q}-{e}_{d}^{"}}{{T}_{q0}^{"}},$`

`$\frac{d{e}_{q}^{\text{'}}}{dt}=\frac{{E}_{fd}-\left({x}_{d}-{x}_{d}^{\text{'}}\right){i}_{d}-{e}_{q}^{\text{'}}}{{T}_{d0}^{\text{'}}},$`

and

`$\frac{d{e}_{q}^{"}}{dt}=\frac{{e}_{q}^{\text{'}}-\left({x}_{d}^{\text{'}}-{x}_{d}^{"}\right){i}_{d}-{e}_{q}^{"}}{{T}_{d0}^{"}},$`

where:

• xd and xq are the d-axis and q-axis synchronous reactances.

• T”d0 and T”q0 are the d-axis and q-axis subtransient open-circuit time constants.

• Efd is the per-unit field voltage using the exciter model nonreciprocal per-unit system.

• x’d is the d-axis transient reactance.

• e’q is the q-axis voltage behind transient reactance.

• T’d0 is the d-axis transient open-circuit time constant.

The rotor torque is defined by

`${T}_{e}={e}_{d}^{"}{i}_{d}+{e}_{q}^{"}{i}_{q}-\left({x}_{d}^{"}-{x}_{q}^{"}\right){i}_{d}{i}_{q}.$`

These defining equations do not describe the parameters you can set in the dialog box. To see their relationship with the equation coefficients, see the book of P. Kundur about understanding, modeling, analyzing, and mitigating power system stability and control problems [1].

### Display Options

You can perform display actions using the Electrical menu on the block context menu.

Right-click the block and, from the Electrical menu, select an option:

• Display Base Values displays the machine per-unit base values in the MATLAB® Command Window.

• Display Associated Base Values displays associated per-unit base values in the MATLAB Command Window.

• Display Associated Initial Conditions displays associated initial conditions in the MATLAB Command Window.

### Variables

The Variables settings allow you to specify the priority and initial target values for block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

For this block, the Variables settings are visible only if, in the Initial Conditions settings, the Initialization option parameter is set to ```Set targets for rotor angle and Park's transform variables```.

## Ports

### Output

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Physical signal vector port associated with the machine per-unit measurements. The vector elements are:

• Field voltage (field circuit base, Efd), pu_fd_Efd

• Field current (field circuit base, Ifd), pu_fd_Ifd

• Electrical torque, pu_torque

• Rotor velocity, pu_velocity

• Stator d-axis voltage, pu_ed

• Stator q-axis voltage, pu_eq

• Stator zero-sequence voltage, pu_e0 — This port is provided to maintain a compatible interface with other machine models. Its value is always zero.

• Stator d-axis current, pu_id

• Stator q-axis current, pu_iq

• Stator zero-sequence current, pu_i0 — This port is provided to maintain a compatible interface with other machine models. Its value is always zero.

• Rotor electrical angle, electrical_angle_out

To connect to this port, use the Synchronous Machine Measurement block.

### Conserving

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Electrical conserving port associated with the field winding positive terminal.

Electrical conserving port associated with the field winding negative terminal.

Mechanical rotational conserving port associated with the machine rotor.

Mechanical rotational conserving port associated with the machine case.

Expandable three-phase port associated with the stator windings.

Electrical conserving port associated with the neutral point of the wye winding configuration. This port is provided to maintain a compatible interface for existing machine models. The voltage and current on this port are ignored.

## Parameters

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### Main

Rated apparent power.

RMS rated line-line voltage.

Nominal electrical frequency at which rated apparent power is quoted.

Number of machine pole pairs.

Block parameterization method. Options are:

• `Fundamental parameters` — Fundamental parameters are visible in the Impedances settings and the Time Constants settings are not visible.

• `Standard parameters` — Standard parameters are visible in the Impedances and the Time Constants settings are visible.

This parameter affects the visibility of the Time Constant settings and the parameters in the Impedances settings.

Field circuit parameterization method. Options are:

• `Field circuit voltage` — Specify the field circuit in terms of voltage.

• `Field circuit current` — Specify the field circuit in terms of current. This method is the default field-circuit parameterization method.

This parameter affects the visibility of the Field circuit voltage and Field circuit current parameters.

Voltage across field circuit which produces rated voltage at machine terminals.

#### Dependencies

This parameter is visible only if the Specify field circuit input required to produce rated terminal voltage at no load by parameter is set to ```Field circuit voltage```.

Current in field circuit which produces rated voltage at machine terminals.

#### Dependencies

This parameter is visible only if the Specify field circuit input required to produce rated terminal voltage at no load by parameter is set to ```Field circuit current```.

Reference point for the rotor angle measurement.

When you select the default value, the rotor d-axis and stator a-phase magnetic axis are aligned when the rotor angle is zero.

The other value you can choose for this parameter is `Angle between the a-phase magnetic axis and the q-axis`. When you select this value, the rotor q-axis and stator a-phase magnetic axis are aligned when the rotor angle is zero.

### Impedances

Unsaturated stator d-axis mutual inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Unsaturated stator q-axis mutual inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator leakage inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator resistance. This parameter must be greater than 0.

Rotor field circuit inductance. This parameter must be greater than 0.

Rotor field circuit resistance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor d-axis damper winding 1 inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor d-axis damper winding 1 resistance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor q-axis damper winding 1 inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor q-axis damper winding 1 resistance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator leakage reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

d-axis synchronous reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

q-axis synchronous reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

d-axis transient reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

d-axis subtransient reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

q-axis subtransient reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

### Time Constants

Select between `Open circuit` and `Short circuit`.

The setting for this parameter affects the visibility of the d-axis time constant parameters.

d-axis transient open-circuit time constant. This parameter must be:

• Greater than 0.

• Greater than d-axis subtransient open-circuit, Td0''.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Open circuit`.

d-axis transient short-circuit time constant. This parameter must be:

• Greater than 0.

• Greater than d-axis subtransient short-circuit, Td''.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Short circuit`.

d-axis subtransient open-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Open circuit`.

d-axis subtransient short-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Short circuit`.

Select between `Open circuit` and `Short circuit`.

The setting for this parameter affects the visibility of the q-axis time constant parameters.

q-axis subtransient open-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify q-axis transient time constant parameter is set to `Open circuit`.

q-axis subtransient short-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify q-axis transient time constant parameter is set to `Short circuit`.

### Initial Conditions

Model for specifying values for certain parameters and variables at the start of simulation. To:

Set an operating point regardless of the connected network, select ```Set real power, reactive power, terminal voltage and terminal phase```.

• Specify the priority and initial target values for block variables before simulation using the Variables settings, select ```Set targets for rotor angle and Park's transform variables```. For more information, see Set Priority and Initial Target for Block Variables.

• Select a bus type and specify the related parameters for a load-flow analysis in the Initial Conditions settings, select ```Set targets for load flow variables```.

#### Dependencies

If you set this parameter to:

• ```Set targets for rotor angle and Park's transform variables``` — The Variables settings become visible.

• ```Set real power, reactive power, terminal voltage, and terminal phase``` — Related parameters become visible.

• ```Set targets for load flow variables``` — Related parameters become visible.

Type of voltage source that the block models.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```Swing bus``` or `PV bus`.

The visibility of Terminal voltage magnitude, Terminal voltage angle, Active power generated, Reactive power generated, Minimum terminal voltage magnitude (pu, Phase search range at terminals, and Phase search range at terminals depend on the value that you choose for this parameter.

Terminal voltage magnitude.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```Swing bus``` or `PV bus`.

Terminal voltage angle.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```Swing bus```.

Active power generated.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PV bus``` or `PQ bus`.

Reactive power generated.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PQ bus```.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PQ bus```.

Vector that defines the phase angle search range.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PV bus``` or `PQ bus`.

Parasitic conductance to the electrical reference.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables```.

## References

[1] Kundur, P. Power System Stability and Control. New York: McGraw Hill, 1993.

[2] Lyshevski, S. E. Electromechanical Systems, Electric Machines and Applied Mechatronics. Boca Raton, FL: CRC Press, 1999.

[3] Pal, M. K. Lecture Notes on Power System Stability. June, 2007.

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