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Model the dynamics of simplified three-phase synchronous machine

Simscape / Electrical / Specialized Power Systems / Fundamental Blocks / Machines

The Simplified Synchronous Machine block models both the electrical and mechanical characteristics of a simple synchronous machine.

The electrical system for each phase consists of a voltage source in series with an RL impedance, which implements the internal impedance of the machine. The value of R can be zero but the value of L must be positive.

The Simplified Synchronous Machine block implements the mechanical system described by

$$\begin{array}{c}\Delta \omega (t)=\frac{1}{2H}{\displaystyle \underset{0}{\overset{t}{\int}}\left(Tm-Te\right)\text{\hspace{0.17em}}}-Kd\Delta \omega (t)dt\\ \omega (t)=\Delta \omega (t)+{\omega}_{0},\end{array}$$

where

Δ*ω* = Speed variation
with respect to speed of operation

*H* =
constant of inertia

*Tm* = mechanical
torque

*Te* = electromagnetic torque

*Kd* = damping factor representing the
effect of damper windings

*ω*(*t*)
= mechanical speed of the rotor

*ω*_{0} =
speed of operation (1 p.u.)

Although the parameters can be entered in either SI units or per unit in the dialog box, the internal calculations are done in per unit. The following block diagram illustrates how the mechanical part of the model is implemented. The model computes a deviation with respect to the speed of operation; not the absolute speed itself.

The Kd damping coefficient simulates the effect of damper windings
normally used in synchronous machines. When the machine is connected
to an infinite network (zero impedance), the variation of machine
power angle delta (δ) resulting from a change of mechanical
power (*P*_{m}) can be approximated
by the following second-order transfer function:

$$\frac{\delta}{{P}_{m}}=\frac{{\omega}_{s}/(2H)}{{s}^{2}+2\zeta {\omega}_{n}s+{\omega}_{n}^{2}},$$

where

δ | Power angle delta: angle of internal voltage |

P | Mechanical power in pu |

ω | Frequency of electromechanical oscillations = $$\sqrt{{\omega}_{s}{P}_{\text{max}}/(2H)}$$ in rad/s |

ζ | Damping ratio = $$\left({K}_{d}/4\right)\sqrt{2/\left({\omega}_{s}H{P}_{\text{max}}\right)}$$ |

ω | Electrical frequency in rad/s |

| Maximum power in pu transmitted through reactance X,
where V_{t}, E,
and X are in pu |

H | Inertia constant(s) |

K | Damping factor (pu_of_torque / pu_of_speed) |

This approximate transfer function, which has been derived by assuming sin(δ) = δ, is valid for small power angles (δ < 30 degrees). It follows from the preceding ζ expression that the Kd value required to obtain a given ζ damping ratio:

$${K}_{d}=4\zeta \sqrt{{\omega}_{s}H{P}_{\text{max}}/2}.$$

In the Machines library you can choose between the SI units or the pu units Simplified Synchronous Machine blocks to specify the electrical and mechanical parameters of the model. These two blocks simulate exactly the same simplified synchronous machine model; the only difference is how you enter the parameter units.

**Connection type**Specify the number of wires used in the three-phase Y connection: three-wire (neutral not accessible) or four-wire (neutral is accessible). Default is

`3-wire Y`

.**Mechanical input**Select the mechanical power applied to the shaft or the rotor speed as a Simulink

^{®}input of the block, or to represent the machine shaft by a Simscape™ rotational mechanical port.Select

`Mechanical power Pm`

(default) to specify a mechanical power input, in W or in pu, and change labeling of the block input to`Pm`

. The machine speed is determined by the machine Inertia J (or inertia constant H for the pu machine) and by the difference between the mechanical torque*Tm*, resulting from the applied mechanical power*Pm*, and the internal electromagnetic torque*Te*. The sign convention for the mechanical power is 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 or in pu, and change labeling of the block input to`w`

. The machine speed is imposed and the mechanical part of the model (inertia constant H) is ignored. Using the speed as the mechanical input allows modeling a mechanical coupling between two machines.The next figure indicates how to model a stiff shaft interconnection in a motor-generator set when friction torque is ignored in machine 2. The speed output of machine 1 (motor) is connected to the speed input of machine 2 (generator), while machine 2 electromagnetic torque output

*Te*is applied to the mechanical torque input*Tm*of machine 1. The Kw factor takes into account speed units of both machines (pu or rad/s) and gear box ratio w2/w1. The KT factor takes into account torque units of both machines (pu or N.m) and machine ratings. Also, because the inertia J2 is ignored in machine 2, J2 referred to machine 1 speed must be added to machine 1 inertia J1.Select

`Mechanical rotational port`

to add to the block a Simscape mechanical rotational port that allows connection of the machine shaft with other Simscape blocks having mechanical rotational ports. The Simulink input representing the mechanical power*Pm*or the speed`w`

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

**Use signal names to identify bus labels**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. Default is cleared.

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.

**Nominal power, line-to-line voltage, and frequency**The nominal apparent power Pn (VA), frequency fn (Hz), and RMS line-to-line voltage Vn (V). Computes nominal torque and converts SI units to pu. Default is

`[187e6 13800 60]`

.**Inertia, damping factor and pairs of poles**The inertia (J in kg.m

^{2}or H in seconds) damping factor (Kd) and number of pairs of poles (p). The damping factor must be specified in (pu of torque)/(pu of speed) in both machine dialog boxes (in pu and in SI). Default is`[3.7 0 20]`

for pu and`[3.895e6 0 20]`

for SI.**Internal impedance**The resistance R (Ω or pu) and reactance L (H or pu) for each phase. Default is

`[0.02 0.3]`

for pu and`[0.0204 0.8104e-3]`

for SI.**Initial conditions**The initial speed deviation (% of nominal), rotor angle (degrees), line current magnitudes (A or pu), and phase angles (degrees). These values are automatically computed by the load flow utility of the Powergui block. Default is

`[ 0,0 0,0,0 0,0,0 ]`

.**Sample time (−1 for inherited)**Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to

`−1`

. Default is`−1`

.

To enable the Advanced tab, set the **Simulation type** parameter of
the powergui block to **Discrete** and, on the
**Preferences** tab, clear the **Automatically handle discrete
solver and Advanced tab solver settings of blocks** parameter.

- Discrete solver model
Specifies the integration method used by the block. The choices are

`Trapezoidal non iterative`

(default),`Trapezoidal robust`

, and`Backward Euler robust`

.When you select the

**Automatically handle discrete solver and Advanced tab solver settings of blocks**parameter in the powergui block, the discrete solver model is automatically set to`Trapezoidal robust`

.`Trapezoidal non iterative`

requires you to add a non-negligible shunt load at the machine terminals to maintain simulation stability, and the simulation may fail to converge and stop when the number of machines increases in the model.`Trapezoidal robust`

and`Backward Euler robust`

allow you to eliminate the need to use parasitic loads. To eliminate the topological errors of the 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.`Trapezoidal robust`

is slightly more accurate than`Backward Euler robust`

, especially when the model is simulated at larger sample times.`Trapezoidal robust`

may produce slight damped numerical oscillations on machine voltage in no-load conditions, while`Backward Euler robust`

prevents oscillations and maintains accuracy.For more information on what method to use in your application, see Simulating Discretized Electrical Systems.

- Sample time (-1 for inherited)
Specifies the sample time used by the block. To inherit the sample time specified in the powergui block, set this parameter to

`-1`

(default).

The load flow parameters define block parameters for use with the Load Flow tool of the Powergui block. These load flow parameters are for model initialization only. They have no impact on the block model or on the simulation performance.

The configuration of the **Load Flow** tab
depends on the option selected for the **Generator type** parameter.

**Generator type**Specify the generator type of the machine.

Select

`swing`

to implement 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.Select

`PV`

(default) to implement a generator controlling its output active power P and voltage magnitude V. P is specified by the**Active power generation P**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**and**Maximum reactive power Qmax**parameters.Select

`PQ`

to implement a generator controlling its output active power P and reactive power Q. P and Q are specified by the**Active power generation P**and**Reactive power generation Q**parameters of the block, respectively.**Active power generation P**Specify the active power that you want generated by the machine, in watts. When the machine operates in motor mode, you specify a negative value. This parameter is available if you specify

**Generator type**as`PV`

or`PQ`

. Default is`0`

.**Reactive power generation Q**Specify the reactive power that you want generated by the machine, in vars. A negative value indicates that the reactive power is absorbed by the machine. This parameter is available only if you specify

**Generator type**as`PQ`

. Default is`0`

.**Minimum reactive power Qmin**This parameter is available only if you specify

**Generator type**as`PV`

. Indicates the 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. Default is`-inf`

.**Maximum reactive power Qmax**This parameter is available only if you specify

**Generator type**as`PV`

. Indicates the 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. Default is`+inf`

.

`Pm`

The mechanical power supplied to the machine, in watts. The input can be a constant signal or it can be connected to the output of the Hydraulic Turbine and Governor block. The frequency of the internal voltage sources depends on the mechanical speed of the machine.

`w`

The alternative block input instead of

`Pm`

(depending on the value of the**Mechanical input**parameter) is the machine speed, in rad/s.`E`

The amplitude of the internal voltages of the block. It can be a constant signal or it can be connected to the output of a voltage regulator. If you use the SI units machine, this input must be in volts phase-to-phase RMS. If you use the pu units machine, it must be in pu.

`m`

The Simulink output of the block is a vector containing measurement signals. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library. Depending on the type of mask that you use, the units are in SI or in pu.

Name

Definition

Units

ias

Stator current is_a

A or pu

ibs

Stator current is_b

A or pu

ics

Stator current is_c

A or pu

va

Terminal voltage Va

V or pu

vb

Terminal voltage Vb

V or pu

vc

Terminal voltage Vc

V or pu

ea

Internal voltage Ea

V or pu

eb

Internal voltage Eb

V or pu

ec

Internal voltage Ec

V or pu

theta

Rotor angle theta

rad

w

Rotor speed wm

rad/s

Pe

Electrical power Pe

W

The electrical system of the Simplified Synchronous Machine block consists solely of a voltage source behind a synchronous reactance and resistance. All the other self- and magnetizing inductances of the armature, field, and damping windings are neglected. The effect of damper windings is approximated by the damping factor Kd. The three voltage sources and RL impedance branches are Y-connected (three wires or four wires). The load might or might not be balanced.

When you use Simplified Synchronous Machine blocks in discrete systems, you might have to use a small parasitic resistive load, connected at the machine terminals, to avoid numerical oscillations. Large sample times require larger loads. The minimum resistive load is proportional to the sample time. 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 simplified 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.

The `power_simplealt`

example
uses the Simplified Synchronous Machine block to represent a 1000
MVA, 315 kV, 60 Hz equivalent source connected to an infinite bus
(Three-Phase Programmable Voltage Source block). The Simplified Synchronous
Machine (SI Units) block is used as a synchronous generator. The internal
resistance and reactance are set respectively to 0.02 pu (1.9845 Ω)
and 0.2 pu (X = 19.845 Ω; L = 0.0526 H). The inertia of the
machine is J = 168,870 kg.m^{2}, corresponding
to an inertia constant H = 3 s. The electrical frequency is ω_{s} =
2*π*60 = 377 rad/s. The machine has two pairs of poles such
that its synchronous speed is 2*π*60/2 = 188.5 rad/s or 1800
rpm.

The **Load Flow** option of the
Powergui has been used to initialize the machine to start simulation
in steady state with the machine generating 500 MW. The required internal
voltage computed by the load flow is 1.0149 pu. Therefore, an internal
voltage E = 315e3*1.0149 = 319,690 Vrms phase-to-phase is specified
in the Constant block connected to the E input. The maximum power
that can be delivered by the machine with a terminal voltage V_{t} =
1.0 pu and an internal voltage E = 1.0149 pu is P_{max} =
V_{t}*E/X = 1.0149/0.2 = 5.0745 pu.

The damping factor Kd is adjusted to obtain a damping ratio ζ = 0.3. The required Kd value is:

$${K}_{d}=4\zeta \sqrt{{\omega}_{s}H{P}_{\text{max}}/2}=64.3$$

Two Fourier blocks measure the power angle δ. This angle is computed as the difference between the phase angle of phase A internal voltage and the phase angle of phase A terminal voltage.

In this example, a step is performed on the mechanical power applied to the shaft. The machine is initially running in steady state with a mechanical power of 505 MW (mechanical power required for an output electrical power of 500 MW, considering the resistive losses). At t = 0.5 s the mechanical power is suddenly increased to 1000 MW.

Run the example and observe the electromechanical transient on the Scope block displaying the power angle δ in degrees, the machine speed in rpm, and the electrical power in MW.

For an initial electrical power Pe = 500 MW (0.5 pu), the load angle δ is 5.65 degrees, which corresponds to the expected value:

$$Pe=\frac{{V}_{t}E\mathrm{sin}\delta}{X}=\frac{1.0\cdot 1.0149\cdot \mathrm{sin}({5.65}^{\circ})}{0.2}=0.5\text{p}\text{.u}\text{.}$$

As the mechanical power is stepped from 0.5 pu to 1.0 pu, the load angle increases and goes through a series of under damped oscillations (damping ratio ζ = 0.3) before stabilizing to its new value of 11.3 degrees. The frequency of the oscillations is given by:

$${f}_{n}=\frac{1}{2\pi}\sqrt{\frac{{\omega}_{s}{P}_{\text{max}}}{2H}}=2.84\text{Hz}\text{.}$$

Excitation System, Hydraulic Turbine and Governor, powergui, Steam Turbine and Governor, Synchronous Machine