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
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.

Variable	Description	Unit
H	Depending on the number of masses, the table of inertia constants is as follows: Generator Low Pressure Turbine B (LPB) Low Pressure Turbine A (LPA) Intermediate Pressure (IP) Turbine High Pressure (HP) Turbine	s
K_d	Depending on the number of masses, the table of absolute damping coefficients is as follows: Generator LPB turbine LPA turbine IP turbine HP turbine	Nm/rad/s
K_ij	Depending on the number of masses, the table of stiffness coefficients is as follows: Generator - LPB turbine LPB turbine - LPA turbine LPA turbine - IP turbine IP turbine - HP turbine	Nm/rad
D	Depending on the number of masses, the table of self damping coefficients is as follows: Generator LPB turbine LPA turbine IP turbine HP turbine	Nm/rad/s
D_ij	Depending on the number of masses, the table of mutual speed deviation damping coefficients is as follows: Generator - LPB turbine LPB turbine - LPA turbine LPA turbine -IP turbine 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: LPB turbine LPA turbine IP turbine HP turbine Note that the sum of F must always be equal to 1.	-

Control Parameters

Variable	Description	Unit	Variable = {Possible Values}
Mechanical control	The mechanical control can be done as follows: Mechanical power: mechanical power from prime mover or load Mechanical torque: mechanical torque from prime mover or load	pu

Variable

Description

Unit

Variable = {Possible Values}

Mechanical control

The mechanical control can be done as follows:

Mechanical power: mechanical power from prime mover or load
Mechanical torque: mechanical torque from prime mover or load

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.

Variable	Description	Unit
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

Variable	Description	Unit	Variable = {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

Name	Description
S	AC side stator connector (supports only 3-phase connections)
N	AC side neutral connector (supports only 1-phase connections)

Inputs

Name	Description	Unit
Tm_i	Mechanical torque
Pm_i	Mechanical power

Outputs

None

Sensors

Name	Description	Unit
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_i	Mechanical torque on the shaft	pu
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
Loading
rad ahead of the d axis.

Base Values for PU Conversion

Base Value	Description
Loading	Base power
Loading	Base stator voltage (peak) for wye connection
Loading	Base stator voltage (peak) for delta connection
Loading	Base stator current (peak)
Loading	Base mechanical torque
Loading	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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