Simplified Synchronous Machine - Model the dynamics of simplified three-phase synchronous machine

Library

Machines

Description

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

where

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. Notice that the model computes a deviation with respect to the speed of operation, and 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:

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 = in rad/s
ζ	Damping ratio
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 (pu) 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 above ζ expression that the Kd value required to obtain a given ζ damping ratio is

Dialog Box and Parameters

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.

Connection type

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

Mechanical input

Allows you to select either the torque applied to the shaft or the rotor speed as the Simulink signal applied to the block's input.

Select Mechanical power Pm to specify a mechanical power input, in W or in pu, and change labeling of the block's 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 the applied mechanical power Pm, and the internal electromagnetic torque Te. The sign convention for the mechanical power is the following: 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's 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 and interfacing with SimMechanics and SimDriveline models.

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 (motor) is connected to the speed input of machine 2 (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 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, as the inertia J2 is ignored in machine 2, J2 referred to machine 1 speed must be added to machine 1 inertia J1.

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). Used to compute nominal torque and convert SI units to pu.

Inertia, damping factor and pairs of poles

The inertia (J in kg.m² or H in seconds) damping factor (Kd) and number of pairs of poles (p). The damping factor should be specified in (pu of torque)/(pu of speed) in both machine dialog boxes (in pu and in SI).

Internal impedance

The resistance R (Ω or pu) and reactance L (H or pu) for each phase.

Initial conditions

The initial speed deviation (% of nominal), rotor angle (degrees), line current magnitudes (A or pu), and phase angles (degrees). These values can be computed by the load flow utility of the Powergui block.

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.

Note These two blocks simulate exactly the same simplified synchronous machine model; the only difference is the way of entering the parameter units.

Inputs and Outputs

Pm

The mechanical power supplied to the machine. 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, these two inputs should be in watts and volts phase-to-phase RMS. If you use the pu units machine, both inputs should be in pu.

m

The Simulink output of the block is a vector containing 12 signals. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library.

Signal	Definition	Units	Symbol
1	Stator current is_a	A or pu	i_sa
2	Stator current is_b	A or pu	i_sb
3	Stator current is_c	A or pu	i_sc
4	Terminal voltage Va	V or pu	v_a
5	Terminal voltage Vb	V or pu	v_b
6	Terminal voltage Vc	V or pu	v_c
7	Internal voltage Ea	V or pu	E_a
8	Internal voltage Eb	V or pu	E_b
9	Internal voltage Ec	V or pu	E_c
10	Rotor angle theta	rad	Θ
11	Rotor speed wm	rad/s	ω
12	Electrical power Pe	W	P_e

Assumptions

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.

Limitations

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, in order 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 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.

Example

The power_simplealt demo 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.m², corresponding to an inertia constant H = 3 s. The electrical frequency is ω_s = 2*π*60/2 = 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 in order 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 in order to obtain a damping ratio ζ = 0.3. According to the formula given in the Description section, the required Kd value is

Two Fourier blocks are used to 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 demo, 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 demo 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:

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