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Model the dynamics of simplified three-phase synchronous machine
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
$$\begin{array}{c}\Delta \omega (t)=\frac{1}{2H}{\displaystyle \underset{0}{\overset{t}{\int}}\left(Tm-Te\right)\text{\hspace{0.17em}}dt}-Kd\Delta \omega (t)\\ \omega (t)=\Delta \omega (t)+{\omega}_{0},\end{array}$$
where
Δω = 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 E with respect to terminal voltage, in radians |
P_{m} | Mechanical power in pu |
ω_{n} | 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_{s} | Electrical frequency in rad/s |
P_{max} | Maximum power in pu transmitted through reactance X at terminal voltage V_{t} and internal voltage E. P_{max} = V_{t}E/X, where V_{t}, E, and X are in pu |
H | Inertia constant(s) |
K_{d} | 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 powerlib 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.
Specify the number of wires used in the three-phase Y connection: either three-wire (neutral not accessible) or four-wire (neutral is accessible).
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 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.
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.
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.
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).
The resistance R (Ω or pu) and reactance L (H or pu) for each phase.
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.
Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to −1.
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.
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 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.
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.
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.
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.
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.
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.
The alternative block input instead of Pm (depending on the value of the Mechanical input parameter) is the machine speed, in rad/s.
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.
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_simplealtpower_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. Simulation results are shown in the following figure.
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:
$${f}_{n}=\frac{1}{2\pi}\sqrt{\frac{{\omega}_{s}{P}_{\text{max}}}{2H}}=2.84\text{Hz}\text{.}$$