# Synchronous Machine Model 2.1 (fundamental)

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

## Library

Machines / Synchronous Machine (Simplified)

## Description

The Synchronous Machine Model 2.1 (fundamental) 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 parameters to define the characteristics of the machine. This block contains a dq Park transformation, so use it only for balanced operation.

## Electrical Defining Equations

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

`${\theta }_{e}\left(t\right)=N{\theta }_{r}\left(t\right),$`
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 [1].

### Display Options

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

Right-click the block and, from the Power Systems 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.

## Parameters

### Main Tab

Rated apparent power

Rated apparent power. The default value is `555e6` `V*A`.

Rated voltage

RMS rated line-line voltage. The default value is `24e3` `V`.

Rated electrical frequency

Nominal electrical frequency at which rated apparent power is quoted. The default value is `60` `Hz`.

Number of pole pairs

Number of machine pole pairs. The default value is `1`.

Specify field circuit input required to produce rated terminal voltage at no load by

Select between `Field circuit voltage` and ```Field circuit current```. The default value is ```Field circuit current```.

Field circuit current

This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to `Field circuit current`. The default value is `1300` `A`.

Field circuit voltage

This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to `Field circuit voltage`. The default value is `92.95` V.

### Impedances Tab

Stator d-axis mutual inductance (unsaturated), Ladu

Unsaturated stator d-axis mutual inductance. The default value is `1.66` pu.

Stator q-axis mutual inductance (unsaturated), Laqu

Unsaturated stator q-axis mutual inductance. The default value is `1.61` pu.

Stator leakage inductance, Ll

Stator leakage inductance. The default value is `0.15` pu.

Stator resistance, Ra

Stator resistance. The default value is `0.003` pu.

Rotor field circuit inductance, Lfd

Rotor field circuit inductance. The default value is `0.165` pu.

Rotor field circuit resistance, Rfd

Rotor field circuit resistance. The default value is `0.0006` pu.

Rotor d-axis damper winding 1 inductance, L1d

Rotor d-axis damper winding 1 inductance. The default value is `0.1713` pu.

Rotor d-axis damper winding 1 resistance, R1d

Rotor d-axis damper winding 1 resistance. The default value is `0.0284` pu.

Rotor q-axis damper winding 1 inductance, L1q

Rotor q-axis damper winding 1 inductance. The default value is `0.1066` pu.

Rotor q-axis damper winding 1 resistance, R1q

Rotor q-axis damper winding 1 resistance. The default value is `0.0650` pu.

### Initial Conditions Tab

Specify initialization by

Select between `Electrical power and voltage output` and ```Mechanical and voltage states```. The default value is ```Electrical power and voltage output```.

Terminal voltage magnitude

Initial RMS line-line voltage. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `24e3` `V`.

Terminal voltage angle

Initial voltage angle. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `0` `deg`.

Terminal active power

Initial active power. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `500e6` `V*A`.

Terminal reactive power

Initial reactive power. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `0` `V*A`.

Initial rotor angle

Initial rotor angle. During steady-state operation, set this parameter to the sum of the load angle and required terminal voltage offset. This parameter is visible only when you set Specify initialization by to ```Mechanical and voltage states```. The default value is `0` `deg`.

Initial voltage behind d-axis subtransient reactance

Initial voltage behind d-axis subtransient reactance. This parameter is visible only when you set Specify initialization by to ```Mechanical and voltage states```. The default value is `0` pu.

Initial voltage behind q-axis transient reactance

Initial voltage behind q-axis transient reactance. This parameter is visible only when you set Specify initialization by to `Mechanical and voltage states`. The default value is `0` pu.

Initial voltage behind q-axis subtransient reactance

Initial voltage behind q-axis subtransient reactance. This parameter is visible only when you set Specify initialization by to ```Mechanical and voltage states```. The default value is` 0` pu.

## Ports

The block has the following ports:

`fd+`

Electrical conserving port corresponding to the field winding positive terminal.

`fd-`

Electrical conserving port corresponding to the field winding negative terminal.

`R`

Mechanical rotational conserving port associated with the machine rotor.

`C`

Mechanical rotational conserving port associated with the machine case.

`pu`

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

• `pu_fd_Efd`

• `pu_fd_Ifd`

• `pu_torque`

• `pu_velocity`

• `pu_ed`

• `pu_eq`

• `pu_e0` — This port is provided to maintain a compatible interface for existing machine models. Its value is always zero.

• `pu_id`

• `pu_iq`

• `pu_i0` — This port is provided to maintain a compatible interface for existing machine models. Its value is always zero.

`~`

Expandable three-phase port associated with the stator windings.

`n`

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.

## References

[1] Kundur, P. Power System Stability and Control. New York, NY: 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. http://www.mkpalconsulting.com/files/stabilitybook.pdf, 2007.