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Permanent Magnet Synchronous Machine (PMSM)


The Permanent Magnet Synchronous Machine (PMSM) component is a Norton equivalent (non-iterative) model. The machine electrical parameters must be specified with fundamental parameters (in per unit). Furthermore, the control of the PMSM can only be done externally through the power or the mechanical torque pin.

The model has the following features:

  • Dampers: up to 2 damper windings on each of the d and q axes.
  • Magnetic saturation: total, independent or only on the d axis.
  • Stator connection: Wye with neutral, Wye grounded, or delta.
  • Shaft: maximum 5 masses.

The PMSM participates in the load flow as a PV node and is represented as a source behind an impedance. The result of the load flow is valid only for a PMSM whose reactances of axis d and q are equal (surface-mounted PMSM).


Mask and Parameters

Electrical Parameters

Variable

Description

Unit

Variable = {Possible Values}

Base power

Nominal power

MVA


Base voltage

Nominal voltage (line to line)

kV


Frequency

Nominal frequency

Hz


Number of poles

Number of poles

-


d-axis damper

Number of damper on d-axis (max 2 allowed)

-


q-axis damper

Number of dampers on q-axis (max 2 allowed)

-


Stator connection

Stator windings connection (wye or delta)

-


Rs

Stator resistance

pu


Xl

Stator leakage reactance

pu


X0

Zero sequence reactance

pu


Xd

d-axis synchronous reactance

pu


Xq

q-axis synchronous reactance

pu


XlD1

D1 damper winding leakage reactance (must be defined when including 1 or 2 d-axes dampers - ignored otherwise)

pu


RD1

D1 damper winding resistance (must be defined when including 1 or 2 d-axes dampers - ignored otherwise)

pu


XlD2

D2 damper winding leakage reactance (must be defined when including 2 d-axes dampers - ignored otherwise)

pu


RD2

D2 damper winding resistance (must be defined when including 2 d-axes dampers - ignored otherwise)

pu


XlQ1

Q1 damper winding leakage reactance (must be defined when including 1 or 2 q-axes dampers - ignored otherwise)

pu


RQ1

Q1 damper winding resistance (must be defined when including 1 or 2 q-axes dampers - ignored otherwise)

pu


XlQ2

Q2 damper winding leakage reactance (must be defined when including 2 q-axes dampers - ignored otherwise)

pu


RQ2

Q2 damper winding resistance (must be defined when including 2 q-axes dampers - ignored otherwise)

pu


Flux magnet

Permanent magnet flux

Wb


Mechanical Data Parameters

Number of mass allows addition of up to 5 masses on the machine shaft.

  • Selection of 1 mass is default, meaning only 1 mass shaft is simulated corresponding to the machine itself.
  • Selection of 2 masses means the machine and the Low Pressure Turbine A (corresponding to mass 2 in the following figure) are simulated.
  • Similarly selection of additional masses adds the next mass shown in the figure below. 

VariableDescriptionUnitVariable = {Possible Values}

H

Depending on the number of masses, the table of inertia constants is as follows:

  1. Generator
  2. Low Pressure Turbine B (LPB)
  3. Low Pressure Turbine A (LPA)
  4. Intermediate Pressure (IP) Turbine
  5. High Pressure (HP) Turbine

s


Kd

Depending on the number of masses, the table of absolute damping coefficients is as follows:

  1. Generator
  2. LPB turbine
  3. LPA turbine
  4. IP turbine
  5. HP turbine

Nm/rad/s


Kij

Depending on the number of masses, the table of stiffness coefficients is as follows:

  1. Generator - LPB turbine
  2. LPB turbine LPA turbine
  3. LPA turbine IP turbine
  4. IP turbine HP turbine

Nm/rad


D

Depending on the number of masses, the table of self damping coefficients is as follows:

  1. Generator
  2. LPB turbine
  3. LPA turbine
  4. IP turbine
  5. HP turbine

Nm/rad/s


Dij

Depending on the number of masses, the table of mutual speed deviation damping coefficients is as follows:

  1. Generator - LPB turbine
  2. LPB turbine LPA turbine
  3. LPA turbine -IP turbine
  4. IP turbine HP turbine

Nm/rad/s


F

Depending on the number of masses, the table of fraction of external torque is defined as the fraction of the total external torque applied to each mass as follows:

  1. LPB turbine
  2. LPA turbine
  3. IP turbine
  4. HP turbine

Note that the sum of F must always be equal to 1.

-

Control Parameters

VariableDescriptionUnitVariable = {Possible Values}

Mechanical control

The mechanical control can be done as follows:

  1. Mechanical power: mechanical power from prime mover or load
  2. Mechanical torque: mechanical torque from prime mover or load

pu


Both mechanical control use torque and and therefore mechanical data from the PMSM. In the mechanical power control, the torque reference is computed internally from the reference power and the actual PMSM speed. As for the mechanical torque control, torque reference is directly provided to the PMSM. 

Saturation Parameters

In the Saturation tab, the user must specify proper saturation curve points if the saturation is set to Enabled. In the case of independent saturation of axes d and q, the two tables represent the positive part of the curve, whereas the negative part is deduced by symmetry. Furthermore, the number of points that can be specified for the d-axis and q-axis curves can be different. On the other hand, in the case of full saturation or d axis only, only the d-axis table is used.

Important: The first point of each saturation curve must not be set to zero. Additionally, the first point corresponds to the linear inductance of the machine in the case of independent saturation or d axis only.


VariableDescriptionUnitVariable = {Possible Values}

Saturation

User can disable or enable Independent, Total or dAxisOnly saturation

-


d-axis saturation curve points

Number of d-axis saturation curve points (max 10 points can be added).

-


Imd

d-axis magnetizing current points of saturation curve.

pu


Fluxmd

d-axis magnetizing flux points of saturation curve.

pu


q-axis saturation curve points

Number of q-axis saturation curve points (max 10 points can be added).

-


Imq

q-axis magnetizing current points of saturation curve.

pu


Fluxmq

q-axis magnetizing flux points of saturation curve.

pu


Load Flow Parameters

VariableDescriptionUnitVariable = {Possible Values}

Voltage

Desired voltage at the terminal of the machine

pu


Active power

Desired active power output at the terminal of the machine

MW



Ports, Inputs, Outputs and Signals Available for Monitoring

Ports

NameDescription

S

AC side stator connector (supports only 3-phase connections)

N

AC side neutral connector (supports only 1-phase connections)

Inputs

NameDescriptionUnit
Tm_iMechanical torque
Pm_iMechanical power

Outputs

None

Sensors

NameDescriptionUnit

Flux0s

Stator zero sequence flux

pu

Fluxds

Stator d-axis flux

pu

Fluxqs

q axis stator flux

pu

Fluxm

Mutual magnetization flux

pu

Fluxmd

d axis mutual magnetization flux

pu

Fluxmq

q axis mutual magnetization flux

pu

I0s

Zero sequence stator current

pu

ID1

d axis damper D1 winding current 

pu

ID2

d axis damper D2 winding current 

pu

IQ1

q axis damper Q1 winding current 

pu

IQ2

q axis damper Q2 winding current 

pu

Ias

Phase A stator current

A

Ibs

Phase B stator current

A

Ics

Phase C stator current

A

Ids

d axis stator current

pu

Iqs

q axis stator current

pu

Im

Magnetization current

pu

Imd

d axis magnetization current

pu

Imq

q axis magnetization current

pu

Pm_i

Mechanical power input

pu

Pmec

Mechanical power SI

Nm

Pmec_pu

Mechanical power pu

pu

Pmec_ss

Calculated mechanical power by load-flow. It is used to initialize the turbine/prime mover block

pu

PowerAngle

Machine load angle

rad

Ps

Active power SI

W

Ps_pu

Active power pu

pu

Qs

Reactive power SI

VAr

RotorAngle

Synchronous machine rotor angle (relative to a synchronous reference frame)

rad

RotorAngle_HP

Mass 5 (HP turbine) rotor angle (relative to a synchronous reference frame)

rad

RotorAngle_IP

Mass 4 (IP turbine) rotor angle (relative to a synchronous reference frame)

rad

RotorAngle_LPA

Mass 3 (LPA turbine) rotor angle (relative to a synchronous reference frame)

rad

RotorAngle_LPB

Mass 2 (LPB turbine) rotor angle (relative to a synchronous reference frame)

rad

Tem

Machine electromagnetic torque SI

Nm

Tem_pu

Machine electromagnetic torque pu

pu

ThetaS

Rotor electrical angle (angle between the axis of phase a and d)

rad

Tm_iMechanical torque on the shaftpu

Tmec

Mechanical torque on the shaft SI

Nm

Tmec_pu

Mechanical torque on the shaft pu

pu

Tmec_HP

Mechanical torque applied to mass (HP turbine)

pu

Tmec_HP_IP

Mechanical torque between the HP and IP turbine masses

pu

Tmec_IP

Mechanical torque applied to mass (IP turbine)

pu

Tmec_IP_LPA

Mechanical torque between the IP and LPA turbine masses

pu

Tmec_LPA

Mechanical torque applied to mass (LPA turbine)

pu

Tmec_LPA_LPB

Mechanical torque between the LPA and LPB turbine masses

pu

Tmec_LPB

Mechanical torque applied to mass (LPB turbine)

pu

Tmec_LPB_GEN

Mechanical torque between the LPB turbine masses and the machine

pu

V0s

Zero sequence stator voltage

pu

Vds

d axis stator voltage

pu

Vqs

q axis stator voltage

pu

Vt

Machine terminal voltage

pu

Wm

Mechanical or electrical speed of the machine

pu

Wm_HP

Mechanical or electrical speed of the HP turbine

pu

Wm_IP

Mechanical or electrical speed of the IP turbine

pu

Wm_LPA

Mechanical or electrical speed of the LPA turbine

pu

Wm_LPB

Mechanical or electrical speed of the LPB turbine

pu

Wrpm

Mechanical speed of the machine

r/min

Additional Information & Model Equations

Additional Information

  • The machine can have in the q axis 2 damper windings, Q1 and Q2. Similarly, it can have in the d axis 2 damper windings, D1 and D2.
  • The rotor reference frame is such that the q axis is  rad ahead of the d axis.

Base Values for PU Conversion

Base ValueDescription

Base power

Base stator voltage (peak) for wye connection

Base stator voltage (peak) for delta connection

Base stator current (peak)

Base mechanical torque

Base flux (peak)

Model Equations

The Park transform is defined as follows:

Where  is the angle between phase a and the d-axis
The equations of the electrical system (p.u. and generating convention) are given by:


Where

: Electric speed (rad / s)

: The flux produced by the permanent magnets (pu)

NB: The flux equations are in per unit. Therefore in this case, any reactance can also be specified as an inductance.

The electromagnetic torque (p.u.) developed by the PMSM is given by:

The mechanical (general) equation of mass i is described as follows:


Where

The mechanical torque developed on the ith mass (Nm)

: The electromagnetic torque developed by the PMSM (Nm)

: The mechanical speed of mass i (rad / s)

: Mechanical synchronous speed (rad / s)

: The mechanical angle of mass i with respect to a frame of reference

: Absolute damping coefficient (Nm / mechanical rad / s)

: Self damping coefficient (Nm / mechanical rad / s)

: Mutual damping coefficient (Nm / mechanical rad / s)

: Angular stiffness (Nm / mechanical rad)

: Inertia constant (s)

NB: When the machine is used in motor mode, the mechanical torque is negative and positive in generator mode.

Limitations

This model ignores hysteresis. Under certain conditions, the PMSM can diverge as the implemented model of the synchronous machine is non-iterative.


References

[1] Dynamic Simulation of Electric Machinery Using Matlab/Simulink. Chee-Mun Ong, Prentice Hall PTR 1997

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