Synchronous Machine pu Standard

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

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

Description

The Synchronous Machine pu Standard block models a synchronous machine in generator or motor mode using standard parameters in pu 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.

For more information on the pu units system, see Per-Unit System of Units.

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:

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

Examples

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Model Sub-Synchronous Resonance in Steam Turbine and Governor on Series-Compensated Network.

The Subsyncronous Resonance in Steam Turbine and Governor System example uses the Synchronous Machine pu Standard block to model sub-synchronous resonance (SSR) in a steam turbine and governor on a series-compensated network.

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 pu. 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, in pu. 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.

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	pu
ibs	Stator current is_b	pu
ics	Stator current is_c	pu
iq	Stator current iq	pu
id	Stator current id	pu
ifd	Field current ifd	pu
ikq1	Damper winding current ikq1	pu
Ikq2	Damper winding current ikq2	pu
ikd	Damper winding current ikd	pu
phimq	Mutual flux phimq	pu
phimd	Mutual flux phimd	pu
vq	Stator voltage vq	pu
vd	Stator voltage vd	pu
lmq	Lmq saturated inductance	pu
lmd	Lmd saturated inductance	pu
dtheta	Rotor angle deviation d_theta	rad
w	Rotor speed wm	pu
Pe	Electrical power Pe	pu
dw	Rotor speed deviation dw	pu
theta	Rotor mechanical angle theta	degree
Te	Electromagnetic torque Te	pu
delta	Load angle delta	degree
Pe0	Output active power Peo	pu
Qe0	Output reactive power Qeo	pu

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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Specify standard parameters, also known as operational parameters, of the synchronous machine (such as steady-state, transient, and/or subtransient reactances and time constants) as inputs to the block. Typically, machine manufacturers provide operational parameters.

In power system analysis software, the synchronous machine equations are commonly solved using the direct-quadrature-zero transformation technique. A data translation technique calculates fundamental parameters from standard parameters [2], [3].

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 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 pu, and expose the Pm port. The machine speed is determined by the inertia constant H 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 pu and expose the w port. The machine speed is imposed and the mechanical part of the model (the inertia constant H) 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 (pu) and the gear box ratio w2/w1. The KT factor takes into account the torque units of both machines (pu) 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, line-to-line voltage, frequency [ Pn(VA) Vn(Vrms) fn(Hz) ] — Nominal power, line-to-line voltage, and frequency
`[6e+04 400 50]` (default) | three-element vector

Total three-phase apparent power (VA), RMS line-to-line voltage (pu), frequency (Hz), and field current (pu).

This parameter is identical to the Nominal power, voltage, frequency, field current [ Pn(VA) Vn(Vrms) fn(Hz) ifn(A) ] parameter of the Synchronous Machine SI Fundamental block, except that you do not specify a nominal field current. This value is not required because the Synchronous Machine pu Standard block does not need the transformation ratio. Because rotor quantities are viewed from the stator, they are converted to pu using the stator base quantities derived from the preceding three nominal parameters.

Reactances [ Xd Xd' Xd'' Xq Xq'' Xl ] (pu) — Reactances
`[2.24 0.17 0.12 1.02 0.13 0.08]` (default) | six-element vector

d-axis synchronous reactance Xd, transient reactance Xd', and subtransient reactance Xd''; the q-axis synchronous reactance Xq and subtransient reactance Xq''; and the leakage reactance Xl (all in pu).

Dependencies

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

Reactances [ Xd Xd' Xd'' Xq Xq' Xq'' Xl ] (pu) — Reactances
`[ 1.81, 0.3, 0.23, 1.76, 0.65, 0.25, 0.15 ]` (default)

d-axis synchronous reactance Xd, transient reactance Xd', and subtransient reactance Xd''; the q-axis synchronous reactance Xq, transient reactance Xq', and subtransient reactance Xq''; and the leakage reactance Xl (all in pu).

Dependencies

To enable this parameter, set Rotor type to Round.

d axis — d-axis time constant
`Short-circuit` (default) | `Open-circuit`

Time constant for the d axis.