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

**Library:**Simscape / Electrical / Specialized Power Systems / Electrical Machines

The Synchronous Machine pu Fundamental block models a synchronous machine in generator or motor mode using fundamental 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 (the
*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.

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:

$$\begin{array}{l}{V}_{d}=-{i}_{d}{R}_{s}-\omega {\psi}_{q}+\frac{d{\psi}_{d}}{dt}\\ {V}_{q}=-{i}_{q}{R}_{s}+\omega {\psi}_{d}+\frac{d{\psi}_{q}}{dt}\\ {V}_{0}=-{i}_{0}{R}_{0}+\frac{d{\psi}_{0}}{dt}\\ {V}_{fd}=\frac{d{\psi}_{fd}}{dt}+{R}_{fd}{i}_{fd}\\ 0=\frac{d{\psi}_{kd}}{dt}+{R}_{kd}{i}_{kd}\\ 0=\frac{d{\psi}_{kq1}}{dt}+{R}_{kq1}{i}_{kq1}\\ 0=\frac{d{\psi}_{kq2}}{dt}+{R}_{kq2}{i}_{kq2}\\ \left[\begin{array}{c}{\psi}_{d}\\ {\psi}_{kd}\\ {\psi}_{fd}\end{array}\right]=\left[\begin{array}{ccc}L{}_{md}+{L}_{l}& {L}_{md}& {L}_{md}\\ {L}_{md}& {L}_{lkd}+\text{}{L}_{f}{}_{1d}+\text{}{L}_{md}& {L}_{f}{}_{1d}+\text{}{L}_{md}\\ {L}_{md}& {L}_{f}{}_{1d}+\text{}{L}_{md}& {L}_{lfd}+\text{}{L}_{f}{}_{1d}+\text{}{L}_{md}\end{array}\right]\text{\hspace{0.17em}}\left[\begin{array}{c}-{i}_{d}\\ {i}_{kd}\\ {i}_{fd}\end{array}\right]\\ \left[\begin{array}{c}{\psi}_{q}\\ \begin{array}{l}{\psi}_{kq1}\\ {\psi}_{kq2}\end{array}\end{array}\right]=\left[\begin{array}{ccc}{L}_{mq}+{L}_{l}& {L}_{mq}& {L}_{mq}\\ {L}_{mq}& {L}_{mq}+{L}_{kq1}& {L}_{mq}\\ {L}_{mq}& {L}_{mq}& {L}_{mq}+{L}_{kq2}\end{array}\right]\left[\begin{array}{l}\begin{array}{c}-{i}_{q}\\ {i}_{kq1}\end{array}\\ {i}_{kq2}\end{array}\right]\end{array}$$

The Synchronous Machine pu Fundamental block allows you to specify the fundamental parameters of a synchronous machine. You enter field and damper winding parameters (resistances, leakage inductances, and mutual inductances) in pu.

This table displays the stator base values.

$${V}_{sbase}={\frac{{V}_{n}\sqrt{2}}{\sqrt{3}}}_{}$$
| Base stator voltage = peak nominal line-to-neutral voltage (V) |

$${I}_{sbase}={\frac{{P}_{n}\sqrt{2}}{{V}_{n}\sqrt{3}}}_{}$$
| Base stator current (A) |

$${Z}_{sbase}={\frac{{V}_{sbase}}{{I}_{sbase}}}_{}=\frac{{V}_{n}{}^{2}}{{P}_{n}}$$
| Base stator impedance (Ω) |

$${\omega}_{base}=2\pi \text{\hspace{0.17em}}{f}_{n}$$
| Base angular frequency (rad/s) |

$${L}_{sbase}={\frac{{Z}_{sbase}}{{\omega}_{base}}}_{}$$
| Base stator inductance (H) |

where:

*P*is the three-phase nominal power (VA)._{n}*V*is the nominal line-to-line voltage (Vrms)._{n}*f*is the nominal frequency (Hz)._{n}*i*is the nominal field current producing nominal stator voltage at no load (A)._{fn}

The stator parameters to specify in the block are:

$${R}_{s\_pu}=\frac{{R}_{s}}{{Z}_{sbase}}$$
| Stator resistance per phase (pu) |

$${L}_{l\_pu}=\frac{{L}_{l}}{{L}_{sbase}}$$
| Stator leakage inductance (pu) |

$${L}_{md\_pu}=\frac{{L}_{md}}{{L}_{sbase}}$$
| Direct-axis magnetizing inductance (pu) |

$${L}_{mq\_pu}=\frac{{L}_{mq}}{{L}_{sbase}}$$
| Quadrature-axis magnetizing inductance (pu) |

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

In the diagram:

*N*and_{s}*N*are the equivalent number of sinusoidally distributed turns of the stator winding and of the field winding, respectively._{f}*R*and_{s}*L*are stator resistance and leakage inductance._{l}*R*′ and_{f}*L*′ are field resistance and leakage inductance._{lfd}

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

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

$${L}_{sfd}=\frac{2}{3}{L}_{md}\frac{{N}_{f}}{{N}_{s}}=\frac{{V}_{sbase}}{{i}_{fn}\text{\hspace{0.17em}}{\omega}_{base}}$$

from which we deduce the transformation ratio:

$$\frac{{N}_{s}}{{N}_{f}}=\frac{2}{3}{L}_{md}\frac{{i}_{fn}\text{\hspace{0.17em}}{\omega}_{base}}{{V}_{sbase}}.$$

The transformation ratio can be also expressed as:

$$\frac{{N}_{s}}{{N}_{f}}=\frac{2}{3}\frac{{I}_{fbase}}{{I}_{sbase}},$$

where *I _{fbase}* is the base field
current. The table shows how to compute field base values.

$${I}_{fbase}={i}_{fn}\text{\hspace{0.17em}}{L}_{md\_pu}{}_{}$$
| Base field current (A) |

$${V}_{fbase}={\frac{{P}_{n}}{{I}_{fbase}}}_{}$$
| Base field voltage (V) |

$${Z}_{fbase}={\frac{{V}_{fbase}}{{I}_{fbase}}}_{}$$
| Base field impedance (Ω) |

$${L}_{fbase}={\frac{{Z}_{fbase}}{{\omega}_{base}}}_{}$$
| Base field inductance (H) |

The actual field parameters are:

*R*′ is field resistance (Ω)_{f}*L*′ is field leakage inductance (H)_{lfd}

The field parameters to enter in the block menu are:

$${R}_{f\_pu}=\frac{{R}_{f}\text{'}}{{Z}_{fbase}}$$
| Field resistance (pu) |

$${L}_{lfd\_pu}=\frac{{L}_{l}{}_{fd}\text{'}}{{L}_{fbase}}$$
| Field leakage inductance (pu) |

The voltage at the **Vf** input of the block is normalized with respect
to nominal field voltage so that a 1 pu 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 the nominal field current so that a 1 pu input
produces *If* = 1 pu.

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.

The `power_SM_Fundamental`

example uses the
Synchronous Machine SI Fundamental block and 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 and 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 [5].

Look at the **Model Properties/PreLoad Fcn** callback section of
the model to see machine specifications and 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 six-cycle fault is applied at t = 0.1 sec. The Scope block 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, in the powergui block, set
**Simulation type** to `Discrete`

. The
model is discretized with a sample time Ts = 50 μs. To obtain a stable model with
such a small load (0.1% of nominal power), in the **Advanced** tab
of each Synchronous Machine block, set **Discrete solver
model** to ```
Trapezoidal iterative (alg.
loop)
```

.

The `power_syncmachine`

example uses the Synchronous Machine block in motor mode. The
simulated system consists of a 150 HP (112 kVA), 762 V industrial-grade synchronous
motor 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 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 60 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] Canay, I.M. "Causes of Discrepancies
on Calculation of Rotor Quantities and Exact Equivalent Diagrams of the Synchronous
Machine." *IEEE Transactions on Power Apparatus and Systems*. PAS-88, no. 7
(1969): 1114–1120.

[2] Moeini, A., et al. “Synchronous
Machine Stability model, an Update to IEEE Std 1110-2002 Data Translation Technique.” *IEEE standards panel sessions*. 2018.

[3] *IEEE Guide for Synchronous Generator Modeling Practices and Applications in
Power System Stability Analyses.*
IEEE Std 1110-2002 (Revision of IEEE Std 1110-1991 [2003]): 1–72.

[4] Krause, P.C. *Analysis of Electric Machinery*. Section 12.5. New York:
McGraw-Hill, 1986.

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

Excitation System | Hydraulic Turbine and Governor | powergui | Simplified Synchronous Machine | Steam Turbine and Governor | Synchronous Machine SI Fundamental | Synchronous Machine pu Standard