Synchronous Machine SI Fundamental

Model dynamics of three-phase round-rotor or salient-pole synchronous machine using fundamental parameters in SI units

Libraries:
Simscape / Electrical / Specialized Power Systems / Electrical Machines

Description

The Synchronous Machine SI Fundamental block models a synchronous machine in generator or motor mode using fundamental parameters in SI units. The operating mode is dictated by the sign of the mechanical power (positive for generator mode or negative for motor mode). The electrical part of the machine is represented by a sixth-order state-space model and the mechanical part is the same as in the Simplified Synchronous Machine block.

The model takes into account the dynamics of the stator, field, and damper windings. The equivalent circuit of the model is represented in the rotor reference frame (qd frame). Stator windings are connected in wye to an internal neutral point. All rotor parameters and electrical quantities are viewed from the stator and identified by primed variables. The subscripts are:

d,q — d- and q-axis quantity
R,s — Rotor and stator quantity
l,m — Leakage and magnetizing inductance
f,k — Field and damper winding quantity

The electrical model of the machine is shown in these diagrams.

Dynamic Model with Unequal Mutual Inductance

The conventional theory of synchronous machine modeling for stability analysis assumes that the mutual inductances between the armature, damper, and field on direct-axis windings are identical. Generally, damper windings are near the air gap, and as a result the flux linking damper circuits are almost equal to the flux linking armature. This hypothesis produces acceptable outcomes for a wide range of stability studies, especially those on the side of the network. However, when it comes to field current studies, there is considerable error. The equivalent circuit dynamic model of a synchronous machine may include an additional inductance representing the difference between field-damper and field-armature mutual inductances on the D-axis [1]. This inductance is typically called Canay inductance. Canay inductance corresponds to the leakage flux, Φ_C, in the following figure and is interpreted as a corrective element in an equivalent model that may have a negative value [2].

The IEEE^® standard 1110-2002 [3] presents the direct and quadratic axes of the synchronous machine dynamic model as shown in the diagrams.

The relevant equations are:

$\begin{array}{l} V_{d} = - i_{d} R_{s} - ω ψ_{q} + \frac{d ψ_{d}}{d t} \\ V_{q} = - i_{q} R_{s} + ω ψ_{d} + \frac{d ψ_{q}}{d t} \\ V_{0} = - i_{0} R_{0} + \frac{d ψ_{0}}{d t} \\ V_{f d} = \frac{d ψ_{f d}}{d t} + R_{f d} i_{f d} \\ 0 = \frac{d ψ_{k d}}{d t} + R_{k d} i_{k d} \\ 0 = \frac{d ψ_{k q 1}}{d t} + R_{k q 1} i_{k q 1} \\ 0 = \frac{d ψ_{k q 2}}{d t} + R_{k q 2} i_{k q 2} \\ [\begin{matrix} ψ_{d} \\ ψ_{k d} \\ ψ_{f d} \end{matrix}] = [\begin{matrix} L {}_{m d}+ L_{l} & L_{m d} & L_{m d} \\ L_{m d} & L_{l k d} + L_{f}_{1 d} + L_{m d} & L_{f}_{1 d} + L_{m d} \\ L_{m d} & L_{f}_{1 d} + L_{m d} & L_{l f d} + L_{f}_{1 d} + L_{m d} \end{matrix}] [\begin{matrix} - i_{d} \\ i_{k d} \\ i_{f d} \end{matrix}] \\ [\begin{matrix} ψ_{q} \\ \begin{array}{l} ψ_{k q 1} \\ ψ_{k q 2} \end{array} \end{matrix}] = [\begin{matrix} L_{m q} + L_{l} & L_{m q} & L_{m q} \\ L_{m q} & L_{m q} + L_{k q 1} & L_{m q} \\ L_{m q} & L_{m q} & L_{m q} + L_{k q 2} \end{matrix}] [\begin{array}{l} \begin{matrix} - i_{q} \\ i_{k q 1} \end{matrix} \\ i_{k q 2} \end{array}] \end{array}$

Base Values, Transformation Ratio, and Rotor Parameters Referred to the Stator

The Synchronous Machine SI Fundamental block allows you to specify the fundamental parameters of a synchronous machine. You enter field and damper winding parameters (resistances, leakage inductances, and mutual inductances) in SI (Ω, H). The RL parameters of the field and damper windings are not the actual field RL values of the machine but the RL values referred to the stator.

Stator Base Values

This table displays the stator base values.

$V_{s b a s e} = {\frac{V_{n} \sqrt{2}}{\sqrt{3}}}_{}$	Base stator voltage = peak nominal line-to-neutral voltage (V)
$I_{s b a s e} = {\frac{P_{n} \sqrt{2}}{V_{n} \sqrt{3}}}_{}$	Base stator current (A)
$Z_{s b a s e} = {\frac{V_{s b a s e}}{I_{s b a s e}}}_{} = \frac{V_{n}^{2}}{P_{n}}$	Base stator impedance (Ω)
$ω_{b a s e} = 2 π f_{n}$	Base angular frequency (rad/s)
$L_{s b a s e} = {\frac{Z_{s b a s e}}{ω_{b a s e}}}_{}$	Base stator inductance (H)

where:

P_n is the three-phase nominal power (VA).
V_n is the nominal line-to-line voltage (Vrms).
f_n is the nominal frequency (Hz).
i_fn is the nominal field current producing nominal stator voltage at no load (A).

The stator parameters to specify in the block are:

R_s	Stator resistance per phase (Ω)
L_l	Stator leakage inductance (H)
L_md	Direct-axis magnetizing inductance viewed from stator (H)
L_mq	Quadrature-axis magnetizing inductance viewed from stator (H)

Stator/Field Transformation Ratio

This figure shows one phase of the stator winding coupled with the field winding.

In the diagram:

N_s and N_f are the equivalent number of sinusoidally distributed turns of the stator winding and of the field winding, respectively.
R_s and L_l are stator resistance and leakage inductance.
R_f′ and L_lfd′ are field resistance and leakage inductance.

When the three stator windings are energized with a three-phase positive-sequence voltage and the field winding is open, the stator magnetizing inductance is L_md. However, when only one phase is energized and the field winding is open, the magnetizing inductance is 2/3 L_md, as shown in the figure.

At no load, when the field winding is rotating at nominal speed and bears the nominal field DC current i_fn, the amplitude of AC voltage (peak value) induced on one phase of the stator is V_sbase.

The maximum mutual inductance between one stator winding and the field winding is obtained when the two windings are aligned and is given by:

$L_{s f d} = \frac{2}{3} L_{m d} \frac{N_{f}}{N_{s}} = \frac{V_{s b a s e}}{i_{f n} ω_{b a s e}}$

from which we deduce the transformation ratio:

$\frac{N_{s}}{N_{f}} = \frac{2}{3} L_{m d} \frac{i_{f n} ω_{b a s e}}{V_{s b a s e}} .$

The transformation ratio can be also expressed as:

$\frac{N_{s}}{N_{f}} = \frac{2}{3} \frac{I_{f b a s e}}{I_{s b a s e}},$

where I_fbase is the base field current. The table shows how to compute field base values.

$I_{f b a s e} = i_{f n} L_{m d_p u}_{}$	Base field current (A)
$V_{f b a s e} = {\frac{P_{n}}{I_{f b a s e}}}_{}$	Base field voltage (V)
$Z_{f b a s e} = {\frac{V_{f b a s e}}{I_{f b a s e}}}_{}$	Base field impedance (Ω)
$L_{f b a s e} = {\frac{Z_{f b a s e}}{ω_{b a s e}}}_{}$	Base field inductance (H)

The actual field parameters are:

R_f′ is field resistance (Ω)
L_lfd′ is field leakage inductance (H)

Field Voltage, Current, and RL Values Referred to the Stator

You can specify field parameters for the field resistance and leakage inductance referred to the stator (R_f, L_lfd).

If the nominal field current i_fn is known, the transformation ratio N_s / N_f is calculated using the same equation as for the stator/field transformation ratio.

According to Krause [4], the field voltage and current referred to the stator (V_f, I_f) are deduced from actual V_f′ , I_f′ values as follows:

$V_{f} = \frac{N_{s}}{N_{f}} V_{f}'$

$I_{f} = \frac{2}{3} \frac{N_{f}}{N_{s}} I_{f}'$

When the actual field resistance R_f′ and leakage inductances L_lfd′ (seen from the rotor) are known, the values referred to the stator are:

$R_{f} = \frac{3}{2} R_{f}' {(\frac{N_{s}}{N_{f}})}^{2}$

$L_{l f d} = \frac{3}{2} L_{l f d}' {(\frac{N_{s}}{N_{f}})}^{2}$

When the nominal field current is not known, and if the pu values of field resistance and leakage inductances are known (R_{f_pu}, L_{lfd_pu}), the corresponding SI values referred to the stator are computed as follows:

$R_{f} = R_{f_p u} \times Z_{s b a s e}$

$L_{l f d} = L_{l f d_p u} \times L_{s b a s e}$

The same conversions are used for RL parameters of damper windings.

Field Voltage and Current

When you specify the nominal field current, the signal at the Vf input port corresponds to the actual field voltage, as in real life. The field current returned by the measurement output also corresponds to the actual field current If.

The nominal field voltage producing nominal stator voltage at no load is given by:

$e_{f n} = R_{f}' \times i_{f n}$

When you do not specify the nominal field current, the signal at the Vf input port corresponds to the actual field voltage referred to the stator. In this case, the nominal field voltage referred to the stator producing nominal stator voltage at no load is:

$e_{f n} (s t a t o r s i d e) = \frac{R_{f_p u}}{L_{m d_p u}} V_{s b a s e} = \frac{R_{f}}{L_{m d} ω_{b a s e}} V_{s b a s e} .$

The field current returned by the measurement output is the field current referred to the stator. The nominal field current referred to the stator is:

$i_{f n} (s t a t o r s i d e) = \frac{I_{s b a s e}}{L_{m d_p u}} = \frac{I_{s b a s e}}{I_{f b a s e}} i_{f n} .$

Examples

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Model 555 MVA, 24 kV, 60 Hz, 3600 rpm Synchronous Generator

The power_SM_Fundamental example uses the Synchronous Machine SI Fundamental block and Synchronous Machine pu Fundamental block to model a 555 MVA, 24 kV, 60 Hz, 3600 rpm synchronous generator. It shows how to specify SI and pu parameters and explains how to compute field and damper rotor winding parameters that are referred to the stator. In addition to the field winding, the round rotor of this machine has three damper windings: one damper in the direct axis and two dampers in the quadrature axis.

Three circuits simulate the same synchronous machine:

Circuit 1: Fundamental parameters are specified in SI. The nominal field current is specified (ifn = 1300 A).
Circuit 2: Fundamental parameters are specified in SI. The nominal field current is not specified (ifn = 0).
Circuit 3: Fundamental parameters are specified in pu.

Machine parameters are taken from an example in Kundur [5].

Look at the Model Properties/PreLoad Fcn callback section of the model to see machine specifications and the computation of stator and field bases, RL rotor parameters referred to the stator, transformation ratio, and nominal field voltage and current.

The machines initially operate in steady state at virtually no load (load = 0.1% of nominal power) with constant field voltage and mechanical power. A phase-to-phase six-cycle fault is applied at t = 0.1 sec. The Scope block shows the comparison between line-to-line AB voltage, phase A stator current, and field current of the three machines.

To simulate the discrete model, in the powergui block, set Simulation type to Discrete. The model is discretized with a sample time Ts = 50 μs. To obtain a stable model with such a small load (0.1% of nominal power), in the Advanced tab of each Synchronous Machine block, set Discrete solver model to Trapezoidal iterative (alg. loop).

Use Synchronous Machine Block in Motor Mode

The power_syncmachine example uses the Synchronous Machine block in motor mode. The simulated system consists of a 150 HP (112 kVA), 762 V industrial-grade synchronous motor connected to a network with a 10 MVA short-circuit level. The machine is initialized for an output electrical power of −50 kW (negative value for motor mode), corresponding to a mechanical power of −48.9 kW. The corresponding values of mechanical power and field voltage are specified by the Pm Step block and the Vf Constant block. The Pm Step block applies a sudden increase of mechanical power from −48.9 kW to −60 kW at time t = 0.1 s.

Run the simulation.

After the load has increased from 48.9 kW to 60 kW at t = 0.1 s, the machine speed oscillates before stabilizing to 1800 rpm. The load angle (angle between terminal voltage and internal voltage) increases from −21 degrees to −53 degrees.

Assumptions and Limitations

In discrete systems, when you set the Discrete solver model parameter of a Synchronous Machine block to Trapezoidal non iterative, you might have to connect a small parasitic resistive load at the machine terminals to avoid numerical oscillations. Large sample times require larger loads. The minimum resistive load is proportional to the sample time. As a rule of thumb, 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 synchronous 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 should be sufficient.

However, if you set the Discrete solver model parameter of a Synchronous Machine block to Trapezoidal iterative (alg. loop), you can use a negligible parasitic load (below 0.1% of nominal power) while preserving numerical stability. This iterative model produces an algebraic loop and results in slower simulation speed.

Ports

Input

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Pm — Mechanical power
scalar

Mechanical power at the machine's shaft, in W. In generator mode, this input can be a positive constant or function or the output of a prime mover block (see the Hydraulic Turbine and Governor or Steam Turbine and Governor blocks). In motor mode, this input is usually a negative constant or function.

Dependencies

To enable this port, set the Mechanical input parameter (Configuration tab) to Mechanical power Pm.

w — Machine speed
scalar

Machine speed, in rad/s.

Dependencies

To enable this port, set the Mechanical input parameter (Configuration tab) to Speed w.

Vf — Field voltage
scalar

Field voltage. This voltage can be supplied by a voltage regulator in generator mode (see the Excitation System block). In motor mode, this value of this input is usually a constant.

The field voltage at this port must be entered in volts DC if the nominal field current Ifn is specified in the Nominal power, voltage, frequency, field current [ Pn(VA) Vn(Vrms) fn(Hz) ifn(A) ] parameter, or in volts referred to stator if Ifn is not specified. To obtain the value of the field voltage producing a 1 pu stator voltage, select the Display nominal field current and voltage producing 1 pu stator voltage check box in the Advanced tab.

Output

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m — Measurement signal
24-element vector

Measurement signals, returned as a 24-element vector. You can demultiplex these signals by using the Bus Selector block.

Name	Definition	Units
ias	Stator current is_a	A
ibs	Stator current is_b	A
ics	Stator current is_c	A
iq	Stator current iq	A
id	Stator current id	A
ifd	Field current ifd	A
ikq1	Damper winding current ikq1	A
Ikq2	Damper winding current ikq2	A
ikd	Damper winding current ikd	A
phimq	Mutual flux phimq	V.s
phimd	Mutual flux phimd	V.s
vq	Stator voltage vq	V
vd	Stator voltage vd	V
lmq	Lmq saturated inductance	H
lmd	Lmd saturated inductance	H
dtheta	Rotor angle deviation d_theta	rad
w	Rotor speed wm	rad/s
Pe	Electrical power Pe	W
dw	Rotor speed deviation dw	pu
theta	Rotor mechanical angle theta	degree
Te	Electromagnetic torque Te	N.m
delta	Load angle delta	degree
Pe0	Output active power Peo	W
Qe0	Output reactive power Qeo	VAR

The angular deviation of the rotor angle theta represents the electrical angle giving the instantaneous position of the rotor with respect to a common reference rotating at synchronous speed. This angle is useful in stability studies to measure the relative positions of the rotors of different machines in a network. The positions of the rotors are then measured with respect to the theta position of a given machine chosen as a reference.

Conserving

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A — Phase A electric terminal
specialized electrical

Specialized electrical conserving port associated with the phase A electrical terminal.

B — Phase B electric terminal
specialized electrical

Specialized electrical conserving port associated with the phase B electrical terminal.

C — Phase C electric terminal
specialized electrical

Specialized electrical conserving port associated with the phase C electrical terminal.

S — Machine rotor shaft
mechanical

Mechanical rotational conserving port associated with the machine rotor shaft.

Dependencies

To enable this port, set the Mechanical input parameter (Configuration tab) to Mechanical rotational port.

Parameters

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In power system analysis software, the synchronous machine equations are commonly solved using the direct-quadrature-zero transformation technique. The RL parameters of field and damper windings are not the actual field RL values of the machine but the RL values referred to the stator. The rotor side parameters are transferred to stator side and normalized before entering the final processing.

Configuration

Preset model — Electrical and mechanical parameters
`No` (default) | `01: 50Hz 400V 8.1kVA 1500RPM` | `02: 50Hz 400V 16kVA 1500RPM` | ...

Set of predetermined electrical and mechanical parameters for various synchronous machine ratings of power (kVA), phase-to-phase voltage (V), frequency (Hz), and rated speed (rpm).

Select one of the preset models to load the corresponding electrical and mechanical parameters. Choices are:

01: 50Hz 400V 8.1kVA 1500RPM
02: 50Hz 400V 16kVA 1500RPM
03: 50Hz 400V 31.3kVA 1500RPM
04: 50Hz 400V 42.5kVA 1500RPM
05: 50Hz 400V 60kVA 1500RPM
06: 50Hz 400V 85kVA 1500RPM
07: 50Hz 400V 250kVA 1500RPM
08: 50Hz 400V 325kVA 1500RPM
09: 50Hz 400V 670kVA 1500RPM
10: 50Hz 400V 910kVA 1500RPM
11: 50Hz 400V 1320kVA 1500RPM
12: 50Hz 400V 2000kVA 1500RPM
13: 60Hz 460V 10.2kVA 1800RPM
14: 60Hz 460V 20kVA 1800RPM
15: 60Hz 460V 37.5kVA 1800RPM
16: 60Hz 460V 52.5kVA 1800RPM
17: 60Hz 460V 72.5kVA 1800RPM
18: 60Hz 460V 100kVA 1800RPM
19: 60Hz 460V 300kVA 1800RPM
20: 60Hz 460V 406kVA 1800RPM
21: 60Hz 460V 800kVA 1800RPM
22: 60Hz 460V 1075kVA 1800RPM
23: 60Hz 460V 1588kVA 1800RPM
24: 60Hz 460V 2500kVA 1800RPM

Select No (default) if you do not want to use a preset model or if you want to modify some of the parameters of a preset model, as described below.

When you select a preset model, the electrical and mechanical parameters in the Parameters tab are dimmed). To start from a preset model and then modify the machine parameters:

Select the preset model for which you want to initialize the parameters.
Change the Preset model parameter to No. This action does not change the machine parameters, but breaks the connection with the preset model.
Modify the machine parameters as you want.

Mechanical input — Mechanical input
`Mechanical power Pm` (default) | `Speed w` | `Mechanical rotational port`

Whether to represent the mechanical power applied to the shaft or the rotor speed as a Simulink^® input of the block, or to represent the machine shaft with a Simscape™ rotational mechanical port.

Select Mechanical power Pm to specify a mechanical power input, in W, and expose the Pm port. The machine speed is determined by the machine inertia J and by the difference between the mechanical torque Tm, which results from the applied mechanical power Pm, and the internal electromagnetic torque Te. When the speed is positive, a positive mechanical power signal indicates generator mode and a negative signal indicates motor mode.

Select Speed w to specify a speed input in rad/s and expose the w port. The machine speed is imposed and the mechanical part of the model (the machine inertia J) is ignored. Using the speed as the mechanical input allows you to model a mechanical coupling between two machines.

The next figure indicates how to model a stiff shaft interconnection in a motor-generator set where both machines are synchronous machines.

The speed output of machine 1 (the motor) is connected to the speed input of machine 2 (the generator). In this figure, friction torque is ignored in machine 2. Therefore, its electromagnetic torque output Te corresponds to the mechanical torque Tm applied to the shaft of machine 1. The corresponding mechanical input power of machine 1 is computed as Pm = Tm*w.The Kw factor takes into account the speed units of both machines (rad/s) and the gear box ratio w2/w1. The KT factor takes into account the torque units of both machines (N.m) and the machine ratings. Also, because inertia J2 is ignored in machine 2, J2 refers to the speed of machine 1 and must be added to machine 1 inertia J1.

Select Mechanical rotational port to expose a Simscape mechanical rotational port, S, that allows you to connect of the machine shaft to another machine shaft or to other Simscape blocks that have mechanical rotational ports.

The figure indicates how to connect an Ideal Torque Source block from the Simscape library to the machine shaft port to represent the machine in motor mode or generator mode when the rotor speed is positive.

Rotor type — Rotor type
`Salient-pole` (default) | `Round`

Specify the rotor type as either Salient-pole or Round (cylindrical). The setting affects the number of rotor circuits in the q-axis (damper windings).

Use signal names to identify bus labels — How to identify bus labels
`off` (default) | `on`

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.

Parameters

Nominal power, voltage, frequency, field current [ Pn(VA) Vn(Vrms) fn(Hz) ifn(A) ] — Nominal power, voltage, frequency, and field current
`[ 187E6 13800 60 1087 ]` (default) | four-element vector

Total three-phase apparent power Pn (VA), RMS line-to-line voltage Vn (V), frequency fn (Hz), and field current ifn (A).

The nominal field current is the current that produces the nominal terminal voltage under no-load conditions. This model was developed with all quantities viewed from the stator, as explained in Krause [4]. The nominal field current makes it possible to compute the transformation ratio of the machine, which allows you to apply the field voltage viewed from the rotor, as in real life. It also allows the field current, which is a variable in the output vector of the model, to be viewed from the rotor.

If the value of the nominal field current is not known, you must enter 0 or leave it blank. Because the transformation ratio cannot be determined in this case, you have to apply the field voltage as viewed from the stator. The field current in the output vector is also viewed from the stator.

Stator [ Rs(ohm) Ll,Lmd,Lmq(H) (Lc(H)) ] — Stator resistance and inductances
`[ 2.9069E-03 3.0892E-04 3.2164E-03 9.7153E-04 ]` (default) | four-element vector

Resistance Rs (Ω), leakage inductance Lls (H), d-axis and q-axis magnetizing inductances Lmd (H) and Lmq (H), and Canay inductance. If the value of the Canay inductance in not known, you must enter 0 or leave it blank.

Field [ Rf'(ohm) Llfd'(H) ] — Field resistance and leakage inductance
`[ 5.9013E-04 3.0712E-04 ]` (default) | two-element vector

Field resistance Rf (Ω) and leakage inductance Llfd (H), both referred to the stator.

Dampers [ Rkd',Llkd' Rkq1',Llkq1' ] (R=ohm,L=H) — Resistance and leakage inductance
`[ 1.1900E-02 4.9076E-04 2.0081E-02 1.0365E-03 ]` (default) | four-element vector

d-axis resistance Rkd (Ω) and leakage inductance Llkd (H), and q-axis resistance Rkq1 (Ω) and leakage inductance Llkq1 (H). All of these values are referred to the stator.

Dependencies

To enable this parameter, set Rotor type to Salient-pole.

Dampers [ Rkd',Llkd' Rkq1',Llkq1' Rkq2',Llkq2' ] (R=ohm,L=H) — Resistance and leakage inductance
`[ 0.0664, 1.387e-3, 0.0292, 6.896e-4, 7.907e-3, 2.477e-3]` (default) | six-element vector

d-axis resistance Rkd (Ω) and leakage inductance Llkd (H), q-axis resistance Rkq1 (Ω) and leakage inductance Llkq1 (H), and q-axis resistance Rkq2 (Ω) and leakage inductance Llkq2 (H). All of these values are referred to the stator.

Dependencies

To enable this parameter, set Rotor type to Round.

Inertia, friction factor, pole pairs [ J(kg.m^2) F(N.m.s) p() ] — Inertia, friction factor, and pole pairs
`[ 3.895e6 0 20 ]` (default) | three-element vector

Inertia coefficient J (kg.m²), friction factor F (N.m.s), and the number of pole pairs p. The friction torque Tf is proportional to the rotor speed ω (Tf = F.ω.) Tf is expressed in N.m, F in N.m.s, and ω in rad/s.

Dependencies

To enable this parameter, set Mechanical input to Mechanical power Pm or Mechanical rotational port.

Pole pairs — Number of pole pairs
`20` (default) | positive integer

Number of pole pairs of the synchronous machine.

Dependencies

To enable this parameter, set Mechanical input to Speed w.

Initial conditions [ dw(%) th(deg) ia,ib,ic(A) pha,phb,phc(deg) Vf(V) ] — Initial conditions
`[ 0 0 0 0 0 0 0 0 70.3192 ]` (default) | nine-element vector

Initial speed deviation Δω (percent of nominal speed), electrical angle of the rotor Θe (degrees), line current magnitudes ia, ib, ic (A) and phase angles pha, phb, phc (degrees), and the initial field voltage Vf (V). You can compute these values automatically by using the Load Flow tool or the Machine Initialization tool of the powergui block.

You can specify the initial field voltage in one of two ways. If you know the nominal field current (first line, last parameter), enter the initial field voltage in volts DC referred to the rotor. Otherwise, enter a zero as the nominal field current and specify the initial field voltage in volts DC referred to the stator. You can determine the nominal field voltage viewed from the stator by selecting the Display nominal field current and voltage producing 1 pu stator voltage check box on the Advanced tab.

Simulate saturation — Simulate magnetic saturation
`off` (default) | `on`

Whether magnetic saturation of the rotor and stator iron is simulated.

Select this check box to provide the matrix of parameters for simulating the saturation.

Clear this check box to not model saturation in your simulation. In this case, the relationship between the ifd and Vt is linear (no saturation).

[ ifd(A) ; Vt(VLL rms)] — No-load saturation curve parameters
`[695.64,774.7,917.5,1001.6,1082.2,1175.9,1293.6,1430.2,1583.7;9660,10623,12243,13063,13757,14437,15180,15890,16567]` (default) | vector

No-load saturation curve parameters. The magnetic saturation of the stator and rotor iron is modeled by a piecewise linear relationship specifying points on the no-load saturation curve. The first row of this matrix contains the values of field currents. The second row contains values of corresponding terminal 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. For saturation, the nominal field current and nominal RMS line-to-line voltage are the base values for the field current and terminal voltage, respectively.

Click Plot to view the no-load saturation curve.

Dependencies

To enable this parameter, select Simulate saturation.

Advanced

To enable the Advanced tab, in the powergui block, set Simulation type to Discrete and clear Automatically handle discrete solver.

Display nominal field current and voltage producing 1 pu stator voltage — Determine nominal field current and voltage
`off` (default) | `on`

Select to determine the nominal field current and voltage viewed from the stator.

For example, without saturation, the figure shows a typical curve. ifn is 1087 A and Vn is 13800 V RMS line-to-line, which is also 11268 V peak line-to-neutral.

The saturation is modeled as a piecewise linear saturation curve by using two lookup tables implementing variations of Lmd and Lmq magnetizing inductances.

The next figure illustrates the good fit graphically (the diamonds are the actual points entered in the dialog box).

In this case, the following values are used:

ifn	1087 A
ifd	[695.64, 774.7, 917.5, 1001.6, 1082.2, 1175.9, 1293.6, 1430.2, 1583.7] A
Vt	[9660, 10623, 12243, 13063, 13757, 14437, 15180, 15890, 16567] V

Sample time (-1 for inherited) — Sample time for block
`-1` (default) | scalar

Sample time used by the block. To inherit the sample time specified in the powergui block, set this parameter to −1.

Discrete solver model — Integration method
`Trapezoidal non iterative` (default) | `Trapezoidal iterative (alg. loop)` | `Trapezoidal robust` | `Backward Euler robust`

Integration method used by the block when the Solver type parameter of the powergui block is set to Discrete.

The Discrete solver model is automatically set to Trapezoidal robust when you select the Automatically handle Discrete solver and Advanced tab solver settings of blocks parameter of the powergui block.

The Trapezoidal non iterative and Trapezoidal iterative (alg. loop) methods are no longer recommended for discretizing the Synchronous Machine SI Fundamental block. Trapezoidal non iterative requires you to add non-negligible shunt loads at the machine terminals to guarantee simulation stability, and Trapezoidal iterative (alg. loop) may fail to converge and cause the simulation to stop when the number of machines increases in the model.

The Trapezoidal robust and Backward Euler robust methods allow you to eliminate the need to use parasitic loads and simulate a machine without loads. To eliminate topological errors of machines connected to an inductive circuit (for example, a circuit breaker connected in series with the machine); the machine models a negligible internal load of 0.01% of nominal power.

The Trapezoidal robust method is slightly more accurate than the Backward Euler robust method, especially when the model is simulated at larger sample times. The Trapezoidal robust method may produce slight damped numerical oscillations on machine voltage at no-load conditions, while the Backward Euler robust method prevents oscillations and maintains good accuracy.

For more information on what method to use in your application, see Simulating Discretized Electrical Systems.

Load Flow

The load flow parameters are used to define block parameters for use with 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.

Generator type — Machine generator type
`PV` (default) | `swing` | `PQ`

Generator type of the machine. Choice are:

swing — Implements a generator controlling magnitude and phase angle of its terminal voltage. The reference voltage magnitude and angle are specified by the Swing bus or PV bus voltage and Swing bus voltage angle parameters of the Load Flow Bus block connected to the machine terminals.
PV — Implements a generator controlling its output active power P and voltage magnitude V. P is specified by the Active power generation P (W) parameter of the block. V is specified by the Swing bus or PV bus voltage parameter of the Load Flow Bus block connected to the machine terminals. You can control the minimum and maximum reactive power generated by the block by using the Minimum reactive power Qmin (var) and Maximum reactive power Qmax (var) parameters.
PQ — Implements a generator controlling its output active power P and reactive power Q. P and Q are specified by the Active power generation P (W) and Reactive power generation Q (var) parameters of the block, respectively.

Active power generation P (W) — Active power generated by machine
`0` (default) | scalar

Active power that you want generated by the machine, in watts. When the machine operates in motor mode, you specify a negative value.

Dependencies

To enable this parameter, set Generator type to PV or PQ.

Reactive power generation Q (var) — Reactive power generated by machine
`0` (default) | scalar

Reactive power that you want generated by the machine, in vars. A negative value indicates that the reactive power is absorbed by the machine.

Dependencies

To enable this parameter, set Generator type to PQ.

Minimum reactive power Qmin (var) — Minimum reactive power generated by machine
`-inf` (default) | scalar

Minimum reactive power that can be generated by the machine while keeping the terminal voltage at its reference value. This reference voltage is specified by the Swing bus or PV bus voltage parameter of the Load Flow Bus block connected to the machine terminals. The default value is -inf, which means that there is no lower limit on the reactive power output.

Dependencies

To enable this parameter, set Generator type to PV.

Maximum reactive power Qmax (var) — Maximum reactive power generated by machine
`inf` (default) | scalar

Maximum reactive power that can be generated by the machine while keeping the terminal voltage at its reference value. This reference voltage is specified by the Swing bus or PV bus voltage parameter of the Load Flow Bus block connected to the machine terminals. The default value is inf, which means that there is no upper limit on the reactive power output.

Dependencies

To enable this parameter, set Generator type to PV.

References

[1] Canay, I.M. "Causes of Discrepancies on Calculation of Rotor Quantities and Exact Equivalent Diagrams of the Synchronous Machine." IEEE Transactions on Power Apparatus and Systems. PAS-88, no. 7 (1969): 1114–1120.

[2] Moeini, A., et al. “Synchronous Machine Stability model, an Update to IEEE Std 1110-2002 Data Translation Technique.” IEEE standards panel sessions. 2018.

[3] IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE Std 1110-2002 (Revision of IEEE Std 1110-1991 [2003]): 1–72.

[4] Krause, P.C. Analysis of Electric Machinery. Section 12.5. New York: McGraw-Hill, 1986.

[5] Kundur, P. Power System Stability and Control. New York, McGraw-Hill, 1994.

Extended Capabilities

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

Version History

Introduced before R2006a

Synchronous Machine SI Fundamental

Description

Dynamic Model with Unequal Mutual Inductance

Base Values, Transformation Ratio, and Rotor Parameters Referred to the Stator

Examples

Model 555 MVA, 24 kV, 60 Hz, 3600 rpm Synchronous Generator

Use Synchronous Machine Block in Motor Mode

Assumptions and Limitations

Ports

Input

Pm — Mechanical power scalar

Dependencies

w — Machine speed scalar

Dependencies

Vf — Field voltage scalar

Output

m — Measurement signal 24-element vector

Conserving

A — Phase A electric terminal specialized electrical

B — Phase B electric terminal specialized electrical

C — Phase C electric terminal specialized electrical

S — Machine rotor shaft mechanical

Dependencies

Parameters

Configuration

Preset model — Electrical and mechanical parameters No (default) | 01: 50Hz 400V 8.1kVA 1500RPM | 02: 50Hz 400V 16kVA 1500RPM | ...

Mechanical input — Mechanical input Mechanical power Pm (default) | Speed w | Mechanical rotational port

Rotor type — Rotor type Salient-pole (default) | Round

Use signal names to identify bus labels — How to identify bus labels off (default) | on

Parameters

Nominal power, voltage, frequency, field current [ Pn(VA) Vn(Vrms) fn(Hz) ifn(A) ] — Nominal power, voltage, frequency, and field current [ 187E6 13800 60 1087 ] (default) | four-element vector

Stator [ Rs(ohm) Ll,Lmd,Lmq(H) (Lc(H)) ] — Stator resistance and inductances [ 2.9069E-03 3.0892E-04 3.2164E-03 9.7153E-04 ] (default) | four-element vector

Field [ Rf'(ohm) Llfd'(H) ] — Field resistance and leakage inductance [ 5.9013E-04 3.0712E-04 ] (default) | two-element vector

Dampers [ Rkd',Llkd' Rkq1',Llkq1' ] (R=ohm,L=H) — Resistance and leakage inductance [ 1.1900E-02 4.9076E-04 2.0081E-02 1.0365E-03 ] (default) | four-element vector

Dependencies

Dampers [ Rkd',Llkd' Rkq1',Llkq1' Rkq2',Llkq2' ] (R=ohm,L=H) — Resistance and leakage inductance [ 0.0664, 1.387e-3, 0.0292, 6.896e-4, 7.907e-3, 2.477e-3] (default) | six-element vector

Dependencies

Inertia, friction factor, pole pairs [ J(kg.m^2) F(N.m.s) p() ] — Inertia, friction factor, and pole pairs [ 3.895e6 0 20 ] (default) | three-element vector

Dependencies

Pole pairs — Number of pole pairs 20 (default) | positive integer

Dependencies

Initial conditions [ dw(%) th(deg) ia,ib,ic(A) pha,phb,phc(deg) Vf(V) ] — Initial conditions [ 0 0 0 0 0 0 0 0 70.3192 ] (default) | nine-element vector

Simulate saturation — Simulate magnetic saturation off (default) | on

[ ifd(A) ; Vt(VLL rms)] — No-load saturation curve parameters [695.64,774.7,917.5,1001.6,1082.2,1175.9,1293.6,1430.2,1583.7;9660,10623,12243,13063,13757,14437,15180,15890,16567] (default) | vector

Dependencies

Advanced

Display nominal field current and voltage producing 1 pu stator voltage — Determine nominal field current and voltage off (default) | on

Sample time (-1 for inherited) — Sample time for block -1 (default) | scalar

Discrete solver model — Integration method Trapezoidal non iterative (default) | Trapezoidal iterative (alg. loop) | Trapezoidal robust | Backward Euler robust

Load Flow

Generator type — Machine generator type PV (default) | swing | PQ

Active power generation P (W) — Active power generated by machine 0 (default) | scalar

Dependencies

Reactive power generation Q (var) — Reactive power generated by machine 0 (default) | scalar

Dependencies

Minimum reactive power Qmin (var) — Minimum reactive power generated by machine -inf (default) | scalar

Dependencies

Maximum reactive power Qmax (var) — Maximum reactive power generated by machine inf (default) | scalar

Dependencies

References

Extended Capabilities

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

Version History

See Also

Pm — Mechanical power
scalar

w — Machine speed
scalar

Vf — Field voltage
scalar

m — Measurement signal
24-element vector

A — Phase A electric terminal
specialized electrical

B — Phase B electric terminal
specialized electrical

C — Phase C electric terminal
specialized electrical

S — Machine rotor shaft
mechanical

Preset model — Electrical and mechanical parameters
`No` (default) | `01: 50Hz 400V 8.1kVA 1500RPM` | `02: 50Hz 400V 16kVA 1500RPM` | ...

Mechanical input — Mechanical input
`Mechanical power Pm` (default) | `Speed w` | `Mechanical rotational port`

Rotor type — Rotor type
`Salient-pole` (default) | `Round`

Use signal names to identify bus labels — How to identify bus labels
`off` (default) | `on`

Nominal power, voltage, frequency, field current [ Pn(VA) Vn(Vrms) fn(Hz) ifn(A) ] — Nominal power, voltage, frequency, and field current
`[ 187E6 13800 60 1087 ]` (default) | four-element vector

Stator [ Rs(ohm) Ll,Lmd,Lmq(H) (Lc(H)) ] — Stator resistance and inductances
`[ 2.9069E-03 3.0892E-04 3.2164E-03 9.7153E-04 ]` (default) | four-element vector

Field [ Rf'(ohm) Llfd'(H) ] — Field resistance and leakage inductance
`[ 5.9013E-04 3.0712E-04 ]` (default) | two-element vector

Dampers [ Rkd',Llkd' Rkq1',Llkq1' ] (R=ohm,L=H) — Resistance and leakage inductance
`[ 1.1900E-02 4.9076E-04 2.0081E-02 1.0365E-03 ]` (default) | four-element vector

Dampers [ Rkd',Llkd' Rkq1',Llkq1' Rkq2',Llkq2' ] (R=ohm,L=H) — Resistance and leakage inductance
`[ 0.0664, 1.387e-3, 0.0292, 6.896e-4, 7.907e-3, 2.477e-3]` (default) | six-element vector

Inertia, friction factor, pole pairs [ J(kg.m^2) F(N.m.s) p() ] — Inertia, friction factor, and pole pairs
`[ 3.895e6 0 20 ]` (default) | three-element vector

Pole pairs — Number of pole pairs
`20` (default) | positive integer

Initial conditions [ dw(%) th(deg) ia,ib,ic(A) pha,phb,phc(deg) Vf(V) ] — Initial conditions
`[ 0 0 0 0 0 0 0 0 70.3192 ]` (default) | nine-element vector

Simulate saturation — Simulate magnetic saturation
`off` (default) | `on`

[ ifd(A) ; Vt(VLL rms)] — No-load saturation curve parameters
`[695.64,774.7,917.5,1001.6,1082.2,1175.9,1293.6,1430.2,1583.7;9660,10623,12243,13063,13757,14437,15180,15890,16567]` (default) | vector

Display nominal field current and voltage producing 1 pu stator voltage — Determine nominal field current and voltage
`off` (default) | `on`

Sample time (-1 for inherited) — Sample time for block
`-1` (default) | scalar

Discrete solver model — Integration method
`Trapezoidal non iterative` (default) | `Trapezoidal iterative (alg. loop)` | `Trapezoidal robust` | `Backward Euler robust`

Generator type — Machine generator type
`PV` (default) | `swing` | `PQ`

Active power generation P (W) — Active power generated by machine
`0` (default) | scalar

Reactive power generation Q (var) — Reactive power generated by machine
`0` (default) | scalar

Minimum reactive power Qmin (var) — Minimum reactive power generated by machine
`-inf` (default) | scalar

Maximum reactive power Qmax (var) — Maximum reactive power generated by machine
`inf` (default) | scalar

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