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Model the dynamics of three-phase round-rotor or salient-pole synchronous machine

The Synchronous Machine block operates in generator or motor modes. The operating mode is dictated by the sign of the mechanical power (positive for generator mode, 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). All rotor parameters and electrical quantities are viewed from the stator. They are identified by primed variables. The subscripts used are defined as follows:

*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

with the following equations.

This model assumes currents flowing into the stator windings. The measured stator currents returned by the Synchronous Machine block (Ia, Ib, Ic, Id, Iq) are the currents flowing out of the machine.

The Synchronous Machine SI Fundamental block and the Synchronous Machine pu Fundamental block allow you to specify the fundamental parameters of a synchronous machine. You enter field and damper windings parameters (resistances, leakage inductances, and mutual inductances) in SI (Ω, H) or in pu. When you enter parameters in SI, 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.

You can compute base values for stator and rotor windings when the parameters are specified in pu. You can also compute the Stator/Field transformation ratio and the field parameters referred to the stator from the actual field parameters (windings RL values, nominal field voltage, and nominal field current).

The stator base values are:

| = base stator voltage = peak nominal line-to-neutral voltage (V) |

| = base stator current (A) |

| = base stator impedance (Ω) |

| = base angular frequency (rad/s) |

| = base stator inductance (H) |

with

P = three-phase
nominal power (VA)_{n} |

V = nominal line-to-line
voltage (Vrms)_{n} |

f = nominal frequency
(Hz)_{n} |

i = nominal
field current producing nominal stator voltage at no load (A)_{fn} |

The stator parameters to specify in the Synchronous Machine SI Fundamental block are:

R = stator resistance
per phase (Ω)_{s} |

L = stator leakage
inductance (H)_{l} |

L = direct-axis
magnetizing inductance viewed from stator (H)_{md} |

L = quadrature-axis
magnetizing inductance viewed from stator (H)_{mq} |

The stator parameters to enter in the Synchronous Machine pu Fundamental block are:

| = stator resistance per phase (pu) |

| = stator leakage inductance (pu) |

| = direct-axis magnetizing inductance (pu) |

| = quadrature-axis magnetizing inductance (pu) |

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

*N _{s}* and

R , _{s}L =
stator resistance and leakage inductance._{l} |

R , _{f}L =
field resistance and leakage inductance._{lfd} |

When the three stator windings are energized with a three-phase
positive-sequence voltage and field winding is open, the stator magnetizing
inductance is *L _{md}*. However,
when only one phase is energized and field winding is open, the magnetizing
inductance is 2/3

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
stator is

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

from which we deduce the transformation ratio:

The transformation ratio can be also expressed as:

where *I _{fbase}* is
the base field current, which is computed as explained below.

| = base field current (A) |

| = base field voltage (V) |

| = base field impedance (Ω) |

| = base field inductance (H) |

The actual field parameters are:

R = field resistance
(Ω)_{f} |

L = field leakage
inductance (H)_{lfd} |

The field parameters to enter in the Synchronous Machine pu Fundamental block menu are:

| = field resistance (pu) |

| = field leakage inductance (pu) |

The field parameters to enter in the Synchronous Machine
SI Fundamental block are the field resistance and leakage
inductance referred to the stator (*R _{f}*′,

If the nominal field current *i _{fn}* is
known, the transformation ratio

According to Krause [1], the field voltage and current referred
to the stator (*V _{f}*′,

When the actual field resistance *R _{f}* and
leakage inductances

When the nominal field current is not known, and if the pu values
of field resistance and leakage inductances are known (*R _{f_pu}*,

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

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

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

When you do not specify the nominal field current, the signal applied to the Vf input 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:

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:

The voltage applied at the Vf input of the Synchronous Machine pu Fundamental block is normalized with respect to nominal field voltage so that a 1pu input produces a 1 pu stator voltage at no load. The If current returned by the measurement output of the block is also normalized with respect to nominal field current so that a 1pu input produces If = 1pu.

In the **powerlib** library you
can choose between three Synchronous Machine blocks
to specify the parameters of the model. They simulate exactly the
same synchronous machine model; the only difference is the way that
you enter the parameter units in the **Parameters** tab.

**Preset model**Provides a 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 in the entries of the dialog box. Select

`No`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 of the dialog box become nonmodifiable (unavailable). To start from a given preset model and then modify machine parameters, do the following:Select the preset model that you want to initialize the parameters.

Change the

**Preset model**parameter value to`No`. This action does not change the machine parameters. By doing so, you just break the connection with the particular preset model.Modify the machine parameters as you want, then click

**Apply**.

**Mechanical input**Allows you to 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, 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.

Select

**Mechanical rotational port**to add to the block a Simscape mechanical rotational port that allows connection of the machine shaft with another machine shaft or 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.

**Rotor type**Specify rotor type:

`Salient-pole`or`Round`(cylindrical). This choice affects the number of rotor circuits in the q-axis (damper windings).**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.

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, voltage, frequency, field current**The 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 nominal terminal voltage under no-load conditions. This model was developed with all quantities viewed from the stator, as explained in Krause [1]. 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. Since 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**The resistance Rs (Ω), leakage inductance Lls (H), and d-axis and q-axis magnetizing inductances Lmd (H) and Lmq (H).

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

**Dampers**The d-axis resistance Rkd' (Ω) and leakage inductance Llkd' (H), the q-axis resistance Rkq1' (Ω) and leakage inductance Llkq1' (H), and (only if round rotor) the q-axis resistance Rkq2' (Ω) and leakage inductance Llkq2' (H). All of these values are referred to the stator.

**Inertia, friction factor, pole pairs**The inertia coefficient J (kg.m

^{2}), 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).**Initial conditions**The initial speed deviation Δω (% 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), in the dialog box, 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 Vfd which produces a nominal Vt**check box on the**Advanced**tab.**Simulate saturation**Specifies whether magnetic saturation of the rotor and stator iron is to be simulated or not.

**[ifd; Vt]**The no-load saturation curve parameters. 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.

You must select the

**Simulate saturation**check box to simulate saturation. Selecting this check box allows you to enter the matrix of parameters for simulating the saturation. If you do not want to model saturation in your simulation, do not select the**Simulate saturation**check box. In this case, the relationship between the ifd and Vt obtained is linear (no saturation).Click

**Plot**to view the no-load saturation curve.

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

This line is identical to the first line of the fundamental parameters in SI dialog box, except that you do not specify a nominal field current. This value is not required here because we do not need the transformation ratio. Since rotor quantities are viewed from the stator, they are converted to pu using the stator base quantities derived from the preceding three nominal parameters.

**Stator; Field; Dampers**Contain exactly the same parameters as in the previous dialog box, but they are expressed here in pu instead of SI units.

**Inertia coefficient, friction factor, pole pairs**The inertia constant H (s), where H is the ratio of energy stored in the rotor at nominal speed over the nominal power of the machine, the friction factor F (pu torque/pu speed), and the number of pole pairs p. The friction torque Tf is proportional to the rotor speed ω (Tf=F.ω, where all quantities are expressed in pu).

**Initial conditions; Simulate saturation; Saturation parameters**The same initial conditions and saturation parameters as in the SI units dialog box, but all values are expressed in pu instead of SI units. For saturation, the nominal field current multiplied by the d-axis magnetizing inductance and nominal RMS line-to-line voltage are the base values for the field current and terminal voltage, respectively.

**Nominal power, line-to-line voltage, and frequency**The same parameters as in the pu Fundamental dialog box.

**Reactances**The d-axis synchronous reactance Xd, transient reactance Xd', and subtransient reactance Xd'', the q-axis synchronous reactance Xq, transient reactance Xq' (only if round rotor), and subtransient reactance Xq'', and finally the leakage reactance Xl (all in pu).

**d-axis time constants; q-axis time constant(s)**Specify the time constants you supply for each axis: either open-circuit or short-circuit.

**Time constants**The d-axis and q-axis time constants (all in s). These values must be consistent with choices made on the two previous lines: d-axis transient open-circuit (Tdo') or short-circuit (Td') time constant, d-axis subtransient open-circuit (Tdo'') or short-circuit (Td'') time constant, q-axis transient open-circuit (Tqo') or short-circuit (Tq') time constant (only if round rotor), q-axis subtransient open-circuit (Tqo'') or short-circuit (Tq'') time constant.

**Stator resistance**The stator resistance Rs (pu).

**Inertia coefficient, friction factor, pole pairs; Initial conditions; Simulate saturation; Saturation parameters**The same parameters as in the pu Fundamental dialog box.

**Display nominal field current and voltage producing 1 pu stator voltage**Select to determine the nominal field current and voltage viewed from the stator. This parameter is visible only for the Synchronous Machine SI Fundamental block.

As an example, without saturation, a typical curve might be as in the following figure. ifn is 1087 A and Vn is 13800 V RMS line-to-line, which is also 11268 V peak line-to-neutral.

Saturation is modeled as a piece-wise linear saturation curve by using two look-up 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 particular 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)**Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to

`−1`.**Discrete solver model**Specifies the integration method used by the block when the

**Solver type**parameter of the Powergui block is set to`Discrete`. The choices are:`Trapezoidal non iterative`,`Trapezoidal iterative (alg. loop)`, and`Forward Euler`.For more info on what method to use in your application, see Simulating Discretized Electrical Systems.

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 and 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`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`.**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`.**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.**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.

The units of inputs and outputs vary according to which dialog box you use to enter the block parameters. If the fundamental parameters in SI units is used, the inputs and outputs are in SI units (except for dw in the vector of internal variables, which is always in pu, and angle Θ, which is always in rad). Otherwise, the inputs and outputs are in pu.

`Pm`The first Simulink input is the mechanical power at the machine's shaft, in Watts or pu. In generating 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 motoring mode, this input is usually a negative constant or function.

`w`The alternative block input instead of

`Pm`(depending on the value of the**Mechanical input**parameter) is the machine speed, in rad/s.`Vf`The second Simulink input of the block is the field voltage. This voltage can be supplied by a voltage regulator in generator mode (see the Excitation System block). It is usually a constant in motor mode.

If you use the model in SI fundamental units, the field voltage Vf must be entered in volts DC if nominal field current Ifn is specified, or in volts referred to stator if Ifn is not specified. To obtain the Vfd producing nominal voltage, select the

**Display nominal field current and voltage producing 1 pu stator voltage**check box in the**Advanced**tab. If you use the model in pu Standard or in pu Fundamental units, Vf must be entered in pu (1 pu of field voltage producing 1 pu of terminal voltage at no load).`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

iq

Stator current iq

A or pu

id

Stator current id

A or pu

ifd

Field current ifd

A or pu

ikq

Damper winding current ikq1

A or pu

Ikq2

Damper winding current ikq2

A or pu

ikd

Damper winding current ikd

A or pu

phimq

Mutual flux phimq

V.s or pu

phimd

Mutual flux phimd

V.s or pu

vq

Stator voltage vq

V or pu

vd

Stator voltage vd

V or pu

lmq

Lmq saturated inductance

H or pu

lmd

Lmd saturated inductance

H or pu

dtheta

Rotor angle deviation d_theta

rad

w

Rotor speed wm

rad/s

Pe

Electrical power Pe

VA or pu

dw

Rotor speed deviation dw

rad/s

theta

Rotor mechanical angle theta

rad

Te

Electromagnetic torque Te

N.m or pu

delta

Load angle delta

rad

Pe0

Output active power Peo

VA or pu

Qe0

Output reactive power Qeo

VAR or pu

In Discrete systems, when you use Synchronous Machine blocks discretized with the trapezoidal non iterative solver or the forward Euler solver, 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. 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 discretize the Synchronous Machine block using the trapezoidal iterative (alg. loop) solver, you can use a negligible parasitic load (below 0.1% of nominal power) while preserving numerical stability. This iterative model producing an algebraic loop results in slower simulation speed.

The `power_SM_fundamental` example shows the
use of the Synchronous Machine SI Fundamental block
and the 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. It also 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 [3].

Look at the **Model Properties/PreLoad Fcn** callback
section of the model to see machine specifications as well as 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 6-cycle fault is applied at t = 0.1 sec. The Scope 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, select **Discrete** simulation
type in the **Configuration parameters** of the Powergui block.
The model is discretized with a sample time Ts = 50 s. To
obtain a stable model with such a small load (0. 1 % of nominal power),
select the **Trapezoidal iterative (alg. Loop)** discrete
solver in the **Advanced** tab of each Synchronous
Machine block.

The `power_syncmachine``power_syncmachine` example
illustrates the use of the Synchronous Machine block
in motor mode. The simulated system consists of an industrial grade
synchronous motor (150 HP (112 kVA), 762 V) 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 in 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 100 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.

[1] Krause, P.C., *Analysis of Electric
Machinery*, McGraw-Hill, 1986, Section 12.5.

[2] Kamwa, I., et al., "Experience
with Computer-Aided Graphical Analysis of Sudden-Short-Circuit Oscillograms
of Large Synchronous Machines," *IEEE ^{®} Transactions
on Energy Conversion*, Vol. 10, No. 3, September 1995.

[3] Kundur, P., *Power System Stability and Control*,
McGraw-Hill, 1994.

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