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Synchronous Machine Model 2.1 (standard)

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

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Machines / Synchronous Machine (Simplified)

Description

The Synchronous Machine Model 2.1 (standard) 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 standard parameters to define the characteristics of the machine. The block converts the standard parameters to fundamental parameters. 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

θe(t)=Nθr(t),

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

Ps=23[cosθecos(θe2π3)cos(θe+2π3)sinθesin(θe2π3)sin(θe+2π3)].

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

ed=ed"Raid+xq"iq

and

eq=eq"xd"idRaiq,

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

    [idiq]=Ps[iaibic].

    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

    [edeq]=Ps[vavbvc].

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

The rotor voltage equation is defined by

efd=Rfdifd,

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

ded"dt=(xqxq")iqed"Tq0",

deq'dt=Efd(xdxd')ideq'Td0',

and

deq"dt=eq'(xd'xd")ideq"Td0",

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

Te=ed"id+eq"iq(xd"xq")idiq.

These defining equations do not describe the short-circuit time constants 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 VA.

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 resistance, Ra

Stator resistance. The default value is 0.003 pu.

Stator leakage reactance, Xl

Stator leakage reactance. The default value is 0.15 pu.

d-axis synchronous reactance, Xd

The d-axis synchronous reactance. The default value is 1.81 pu.

q-axis synchronous reactance, Xq

The q-axis synchronous reactance. The default value is 1.76 pu.

d-axis transient reactance, Xd'

The d-axis transient reactance. The default value is 0.3 pu.

d-axis subtransient reactance, Xd''

The d-axis subtransient reactance. The default value is 0.23 pu.

q-axis subtransient reactance, Xq''

The q-axis subtransient reactance. The default value is 0.25 pu.

Time Constants Tab

Specify d-axis transient time constant

Select between Open-circuit value and Short-circuit value. The default value is Open-circuit value.

d-axis transient open-circuit, Td0'

The d-axis transient open-circuit time constant. This parameter is visible only when Specify d-axis transient time constant is set to Open-circuit value. The default value is 8 s.

d-axis transient short-circuit, Td'

The d-axis transient short-circuit time constant. This parameter is visible only when Specify d-axis transient time constant is set to Short-circuit value. The default value is 1.326 s.

Specify d-axis subtransient time constant

Select between Open-circuit value and Short-circuit value. The default value is Open-circuit value.

d-axis subtransient open-circuit, Td0''

The d-axis subtransient open-circuit time constant. This parameter is visible only when Specify d-axis subtransient time constant is set to Open-circuit value. The default value is 0.03 s.

d-axis subtransient short-circuit, Td''

The d-axis subtransient short-circuit time constant. This parameter is visible only when Specify d-axis subtransient time constant is set to Short-circuit value. The default value is 0.023 s.

Specify q-axis subtransient time constant

Select between Open-circuit value and Short-circuit value. The default value is Open-circuit value.

q-axis subtransient open-circuit, Tq0''

The q-axis subtransient open-circuit time constant. This parameter is visible only when Specify q-axis subtransient time constant is set to Open-circuit value. The default value is 0.07 s.

q-axis subtransient short-circuit, Tq''

The q-axis subtransient short-circuit time constant. This parameter is visible only when Specify q-axis subtransient time constant is set to Short-circuit value. The default value is 0.0269 s.

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.

Introduced in R2015a

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