Synchronous Machine Model 2.1

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

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Libraries:
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 to initialize a 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

$${\theta }_e(t)=N{\theta }_r(t),$$

where:

• θ_e is the electrical angle.

• N is the number of pole pairs.

• θ_r 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

$$P_s=\frac{2}{3}\left[\begin{array}{ccc}\cos {\theta }_e& \cos ({\theta }_e-\frac{2\pi }{3})& \cos ({\theta }_e+\frac{2\pi }{3})\\ -\sin {\theta }_e& -\sin ({\theta }_e-\frac{2\pi }{3})& -\sin ({\theta }_e+\frac{2\pi }{3})\end{array}\right].$$

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.

• R_a is the stator resistance.

• i_d and i_q are the d-axis and q-axis stator currents, defined by

  $$\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]=P_s\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right].$$

  i_a, i_b, and i_c are the stator currents flowing from port ~ to port n.

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

• e_d and e_q are the d-axis and q-axis stator voltages, defined by

  $$\left[\begin{array}{c}{e}_d\\ e_q\end{array}\right]=P_s\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right].$$

  v_a, v_b, and v_c 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:

• R_fd is the resistance of rotor field circuit.

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

• e_fd 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:

• x_d and x_q 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.

• E_fd 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.

Model Thermal Effects

You can expose thermal ports to model the effects of generated heat and machine temperature. To expose the thermal ports, set the Modeling option parameter to either:

• No thermal port — The block does not contain thermal ports.

• Show thermal port — The block contains multiple thermal conserving ports.

For more information about using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

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

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Ports

Output

pu — Machine per-unit measurements

physical

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

fd+ — Field winding positive terminal

electrical

Electrical conserving port associated with the field winding positive terminal.

fd- — Field winding negative terminal

electrical

Electrical conserving port associated with the field winding negative terminal.

R — Machine rotor

mechanical

Mechanical rotational conserving port associated with the machine rotor.

C — Machine case

mechanical

Mechanical rotational conserving port associated with the machine case.

~ — Stator windings

electrical

Expandable three-phase port associated with the stator windings.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports.

n — Neutral phase

electrical

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.

a — a-phase

electrical

Electrical conserving port associated with a-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

b — b-phase

electrical

Electrical conserving port associated with b-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

c — c-phase

electrical

Electrical conserving port associated with c-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

HA — a-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding a.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HB — b-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding b.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HC — c-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding c.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HR — Rotor thermal port

thermal

Thermal conserving port associated with the rotor.

Dependencies

To enable this port, set Modeling option to Show thermal port.

Parameters

Modeling option — Option to enable thermal ports

No thermal port (default) | Show thermal port

Option to enable the thermal ports of the block and model the effects of generated heat and machine temperature.

Main

Electrical connection — Electrical connection

Composite three-phase ports (default) | Expanded three-phase ports

Whether to have composite or expanded three-phase ports.

Rated apparent power — Rated apparent power

555e6 V*A (default)

Rated apparent power.

Rated voltage — RMS rated line-line voltage

24e3 V (default)

RMS rated line-line voltage.

Rated electrical frequency — Nominal electrical frequency

60 Hz (default)

Nominal electrical frequency at which rated apparent power is quoted.

Number of pole pairs — Number of pole pairs

1 (default)

Number of machine pole pairs.

Specify parameterization by — Block parameterization

Fundamental parameters (default) | Standard parameters

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.

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

Field circuit current (default) | Field circuit voltage

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.

Field circuit voltage — Field circuit voltage

92.95 V (default)

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.

Field circuit current — Field circuit current

1300 A (default)

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.

Rotor angle definition — Reference point for the rotor angle measurement

Angle between the a-phase magnetic axis and the d-axis (default) | Angle between the a-phase magnetic axis and the q-axis

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

Stator d-axis mutual inductance (unsaturated), Ladu — Stator d-axis mutual inductance (unsaturated)

1.66 (default) | positive scalar

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.

Stator q-axis mutual inductance (unsaturated), Laqu — Stator q-axis mutual inductance (unsaturated)

1.61 (default) | positive scalar

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, Ll — Stator leakage inductance

0.15 (default) | positive scalar

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, Ra — Stator resistance

0.003 (default) | positive scalar

Stator resistance. This parameter must be greater than 0.

Rotor field circuit inductance, Lfd — Rotor field circuit inductance

0.165 (default) | positive scalar

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

Rotor field circuit resistance, Rfd — Rotor field circuit resistance

0.0006 (default) | positive scalar

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, L1d — Rotor d-axis damper winding 1 inductance

0.1713 (default) | positive scalar

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, R1d — Rotor d-axis damper winding 1 resistance

0.0284 (default) | positive scalar

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, L1q — Rotor q-axis damper winding 1 inductance

0.1066 (default) | positive scalar

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, R1q — Rotor q-axis damper winding 1 resistance

0.0650 (default) | positive scalar

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, Xl — Stator leakage reactance

0.15 (default) | positive scalar

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, Xd — d-axis synchronous reactance

1.81 (default) | positive scalar

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, Xq — q-axis synchronous reactance

1.76 (default) | positive scalar

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, Xd' — d-axis transient reactance

0.3 (default) | positive scalar

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, Xd'' — d-axis subtransient reactance

0.23 (default) | positive scalar

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, Xq'' — q-axis subtransient reactance

0.25 (default) | positive scalar

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

Specify d-axis time constant — Specify d-axis time constant

Open circuit (default) | Short circuit

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, Td0' — d-axis transient open-circuit

8 s (default) | positive scalar

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, Td' — d-axis transient short-circuit

1.3260 s (default) | positive scalar

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, Td0'' — d-axis subtransient open-circuit

0.03 s (default) | positive scalar

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, Td'' — d-axis subtransient short-circuit

0.0230 s (default) | positive scalar

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.

Specify q-axis time constant — Specify q-axis time constant

Open circuit (default) | 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, Tq0'' — q-axis subtransient open-circuit

0.07 s (default) | positive scalar

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, Tq'' — q-axis subtransient short-circuit

0.0269 s (default) | positive scalar

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

Initialization option — Initialization option

Set real power, reactive power, terminal voltage, and terminal phase (default) | Set targets for rotor angle and Park's transform variables | Set targets for load flow variables

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.

Source type — Voltage source model

Swing bus (default) | Voltage source with series impedance | PV bus | PQ bus

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 — Terminal voltage magnitude

24e3 V (default)

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 — Terminal voltage angle

0 deg (default)

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 — Active power generated

500e6 V*A (default)

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 — Reactive power generated

0 V*A (default)

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.

Minimum terminal voltage magnitude (pu) — Minimum terminal voltage magnitude

0.95 (default)

Per-unit minimum steady-state voltage magnitude.

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.

Phase search range at terminals — Phase search range at terminals

[-inf, inf] deg (default)

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 electrical reference — Parasitic conductance to electrical reference

1e-6 S (default)

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.

Thermal

To enable these parameters, set Modeling option to Show thermal port.

Measurement temperature — Rated temperature

298.15 k (default) | real scalar

Temperature for which motor parameters are quoted.

Resistance temperature coefficient — Temperature coefficient

3.93e-3 1/K (default) | positive scalar

Coefficient α in the equation relating resistance to temperature for all three windings, as described in Thermal Model for Actuator Blocks. The default value, 3.93e-3 1/K, is for copper.

Thermal mass for each stator winding — Winding thermal mass

100 J/K (default) | positive scalar

Thermal mass value for each stator winding. The thermal mass is the energy required to raise the temperature by one degree.

Rotor thermal mass — Rotor thermal mass

200 J/K (default) | positive scalar

Thermal mass of the rotor. The thermal mass is the energy required to raise the temperature of the rotor by one degree.

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2015a

R2021b: Electrical connection ports update

From R2021b forward, to switch between composite and expanded ports, set the Electrical connection parameter to either Composite three-phase ports or Expanded three-phase ports. Also, the simlog format is identical for Composite three-phase ports and Expanded three-phase ports.

As a result of these changes, inside a model saved in an earlier release, review the Electrical connection parameter of this block.