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The Synchronous Machine (SM) component is a Norton equivalent (non-iterative) model. The machine electrical parameters must be specified with standard parameters (in per unit). Furthermore, the control of the SM can only be done externally through the mechanical power or the mechanical torque and the excitation input.
The model has the following features:
- Dampers: up to 3 damper winding, one on d-axis and two on q-axis
- Magnetic saturation: total, independent or only on the d axis.
- Stator connection: Wye with neutral, Wye grounded, or delta.
- Shaft: maximum 6 masses including the mass of the exciter.
The synchronous machine participates in the load flow as a source behind an impedance. The machine may be a reference node (swing bus), PV node or PQ node.
Mask and Parameters
Electrical Parameters
Parameter | 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 dampers on d-axis | - | |
q-axis damper | Number of dampers on q-axis | - | |
Stator connection | Stator windings connection (wye or delta) | - | |
| Field current | Field current which will produced 1 pu voltage on the air gap line used to calculate the pu value of current values input to the saturation table (not available in the current version) | A | |
Rs | Stator resistance | pu | |
Xl | Stator leakage reactance | pu | |
X0 | Zero sequence reactance | pu | |
Xc | Canay reactance | pu | |
Xd | d-axis synchronous reactance | pu | |
Xq | q-axis synchronous reactance | pu | |
| Xpd | q-axis transient reactance (must be defined when including any number of d-axis dampers - ignored otherwise) | pu | |
Xpq | q-axis transient reactance (must be defined when including any number of q-axis dampers - ignored otherwise) | pu | |
Xppd | d-axis subtransient reactance (must be defined when including 1 d-axes dampers - ignored otherwise) | pu | |
Xppq | q-axis subtransient reactance (must be defined when including 2 q-axes dampers - ignored otherwise) | pu | |
Tpd0 | d-axis open-circuit transient constant time (must be defined when including any number of d-axis dampers - ignored otherwise) | s | |
Tpq0 | q-axis open-circuit transient constant time (must be defined when including any number of q-axis dampers - ignored otherwise) | s | |
Tppd0 | d-axis open-circuit subtransient time constant (must be defined when including 1 d-axes dampers - ignored otherwise) | s | |
Tppq0 | d-axis open-circuit subtransient time constant (must be defined when including 2 q-axes dampers - ignored otherwise) | s |
Mechanical Data Parameters
Parameter | Description | Unit | Variable = {Possible Values} |
|---|---|---|---|
Number of mass | Number of mass | - | |
H | Depending on the number of mass, the table of inertia constants is as follows:
| s | |
Kd | Depending on the number of mass, the table of absolute damping coefficients is as follows:
| Nm/rad/s | |
Kij | Depending on the number of mass, the table of stiffness coefficients is as follows:
| Nm/rad | |
D | Depending on the number of mass, the table of self damping coefficients is as follows:
| Nm/rad/s | |
Dij | Depending on the number of mass, the table of mutual speed deviation damping coefficients is as follows:
| 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:
Note that the sum of F must always be equal to 1. | - | |
Exciter Type | Type of exciter (rotating or static) | - | |
H (Exciter) | Inertia constant | s | |
Kd (Exciter) | Absolute damping coefficient | Nm/rad/s | |
D (Exciter) | Self damping coefficient | Nm/rad/s | |
D(Generator-Exciter) | Mutual deviation speed damping coefficient | Nm/rad/s | |
K(Generator-Exciter) | Stiffness coefficient | Nm/rad |
Control Parameters
Parameter | Description | Unit | Variable = {Possible Values} |
Excitation control | The excitation is controlled as follows:
| pu | |
Mechanical control | The mechanical control can be done as follows:
| pu |
Saturation Parameters
Parameter | Description | Unit | Variable = {Possible Values} |
Saturation | User can disable or enable independent, total or only d-axis 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 |
Note: The magnetizing current of axis d or q are defined in p.u of the exciter i.e. base of the rotor current. For example if a field current of 1000 A produces an air gap flux of 1 pu then this current is equal to 1 p.u. A 1500 A current producing an air gap flux of 1.1 pu then this current is
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Load Flow Parameters
Parameter | Description | Unit | Variable = {Possible Values} |
Type of bus | The bus can be:
| - | |
Voltage | Desired voltage at the terminal of the machine | pu | |
Angle | Desired voltage angle at the terminal of the machine | Deg | |
Active power | Desired active power output at the terminal of the machine | MW | |
Reactive power | Desired reactive power output at the terminal of the machine | MVAr | |
Minimum power reactive | Minimum allowed reactive output at the terminal of the machine | MVAr | |
Maximum power reactive | Maximum allowed reactive output at the terminal of the machine | MVAr |
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 |
|---|---|---|
| Vfd_i | Speed independent field Voltage | |
| Tm_i | Mechanical torque | |
| Pm_i | Mechanical power | |
| Efd_i | Speed dependent field voltage |
Outputs
None
Sensors
Last name | Description | Unit |
|---|---|---|
Efd | Speed-dependent field voltage | pu |
Efd_i | Speed-dependent field voltage of the exciter | pu |
Efd_ss | The exciter’s field voltage calculated during load flow. It is used to initialize the blocks of the exciter. ss: steady state. | pu |
Flux0s | Zero-sequence Stator flux | pu |
Fluxds | d-axis stator flux | pu |
Fluxm | Mutual Magnetization flux | pu |
Fluxmd | Mutual magnetization flux of d-axis | pu |
Fluxmq | Mutual magnetization flux of q-axis | pu |
Fluxqs | q-axis stator flux | pu |
I0s | Zero-sequence stator current | pu |
ID | d-axis damper current | pu |
IQ1 | Q1 axis damper current | pu |
IQ2 | Q2 axis damper current | pu |
Ias | stator current of phase A | A |
Ibs | stator current of phase B | A |
Ics | stator current of phase C | A |
Ids | d-axis stator current | pu |
If | field current | pu |
Im | Magnetizing current | pu |
Imd | d-axis magnetizing current | pu |
Imq | q-axis magnetizing current | pu |
Iqs | q-axis stator current | pu |
Pm_i | Mechanical power of the turbine | pu |
Pmec | Mechanical power of the turbine | Nm |
Pmec_pu | Mechanical power of the turbine | pu |
Pmec_ss | Mechanical power of the turbine calculated during load flow. It is used to initialize the blocks of the turbine | pu |
PowerAngle | The load angle of the machine | rad |
Ps | active power | W |
Ps_pu | active power | pu |
Qs | reactive power | VAr |
RotorAngle | Rotor angle (relative to synchronous reference frame) of the synchronous machine | rad |
RotorAngle_EXC | Rotor angle (relative to synchronous reference frame) of the mass (exciter) | rad |
RotorAngle_HP | Rotor angle (relative to a synchronous reference frame) of the mass (HP turbine) | rad |
RotorAngle_IP | Rotor angle (relative to a synchronous reference frame) of the mass (IP turbine) | rad |
RotorAngle_LPA | Rotor angle (relative to a synchronous reference frame) of the mass (LPA turbine) | rad |
RotorAngle_LPB | Rotor angle (relative to a synchronous reference frame) of the mass (turbine LPB) | rad |
Tem | electromagnetic torque of the synchronous machine | Nm |
Tem_pu | electromagnetic torque of the synchronous machine | pu |
Texc | electromagnetic torque of the exciter | pu |
ThetaS | Electrical angle of rotor (angle between the axis of phase A and d) | rad |
| Tm_i | Mechanical torque of the machine | Nm |
Tmec | mechanical torque of the turbine | Nm |
Tmec_GEN_EXC | mechanical torque between the synchronous machine mass and the exciter | pu |
Tmec_HP | mechanical torque applied to the mass (HP turbine) | pu |
Tmec_HP_IP | mechanical torque between HP and IP turbine masses | pu |
Tmec_IP | mechanical torque applied to the mass (IP turbine) | pu |
Tmec_IP_LPA | mechanical torque between the turbine masses IP and LPA | pu |
Tmec_LPA | mechanical torque applied to the mass (LPA turbine) | pu |
Tmec_LPA_LPB | mechanical torque between the turbine masses LPA and LPB | pu |
Tmec_LPB | mechanical torque applied to the mass (turbine LPB) | pu |
Tmec_LPB_GEN | mechanical torque between the masses LPB turbine and the synchronous machine | pu |
Tmec_pu | mechanical torque of the turbine | pu |
V0s | zero sequence stator voltage | pu |
Vds | d-axis stator voltage | pu |
Vqs | q axis stator voltage | pu |
Vfd | Speed independent field Voltage | pu |
Vfd_i | Speed independent field voltage of the exciter | pu |
Vt | Terminal voltage of the machine | pu |
Wm | mechanical or electrical speed of the machine | pu |
Wm_EXC | mechanical or electrical speed of the exciter | pu |
Wm_HP | mechanical or electrical speed of the turbine HP | pu |
Wm_IP | mechanical or electrical speed of the IP turbine | pu |
Wm_LPA | mechanical or electrical speed of the turbine LPA | pu |
Wm_LPB | mechanical or electrical speed of the turbine LPB | pu |
Wrpm | Mechanical speed of the machine | r / min |
Additional Information & Model Equations
Additional Information
In the q-axis, the machine has 2 damper windings Q1, Q2. In the d-axis (the axis of the magnetic field), one d-axis damper winding. The rotor reference is such that the q-axis leads the d-axis.
Base Values for PU Conversion
| Base Value | Description | ||||
|---|---|---|---|---|---|
| Base power | ||||
| Base stator voltage (peak) for wye connection | ||||
| Base stator voltage (peak) for delta connection | ||||
| Base stator current (peak) | ||||
| Base rotor current | ||||
| Base mechanical torque | ||||
| Base flux (peak) |
Model Equations
Park's transformation is defined as follows:
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T=\frac{2}{3}\left(\begin{array}{ccc}
{\cos \theta_{s}} & {\cos \left(\theta_{s}-\frac{2 \pi}{3}\right)} & {\cos \left(\theta_{s}+\frac{2 \pi}{3}\right)} \\
{-\sin \theta_{s}} & {-\sin \left(\theta_{s}-\frac{2 \pi}{3}\right)} & {-\sin \left(\theta_{s}+\frac{2 \pi}{3}\right)} \\
{\frac{1}{2}} & {\frac{1}{2}} & {\frac{1}{2}}
\end{array}\right) |
Where
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The equations of the electrical system (p.u. with generating convention) are given by:
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v_{q}=-r_{s} i_{q}+\frac{\omega_{r}}{\omega_{s}} \psi_{d}+\frac{1}{\omega_{s}} \frac{d \psi_{q}}{d t} |
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v_{d}=-r_{s} i_{d}-\frac{\omega_{r}}{\omega_{s}} \psi_{q}+\frac{1}{\omega_{s}} \frac{d \psi_{d}}{d t} |
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v_{0}=-r_{s} i_{0}+\frac{1}{\omega_{s}} \frac{d \psi_{0}}{d t} |
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v_{f}=r_{f} i_{f}+\frac{1}{\omega_{s}} \frac{d \psi_{f}}{d t} |
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0=r_{D} i_{D}+\frac{1}{\omega_{s}} \frac{d \psi_{D}}{d t} |
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0=r_{Q 1} i_{Q 1}+\frac{1}{\omega_{s}} \frac{d \psi_{Q 1}}{d t} |
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0=r_{Q 2} i_{Q 2}+\frac{1}{\omega_{s}} \frac{d \psi_{Q 2}}{d t} |
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\psi_{q}=-x_{q} i_{q}+x_{m q} i_{Q 1}+x_{m q} i_{Q 2} |
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\psi_{d}=-x_{d} i_{d}+x_{m d}i_{D} +x_{m d} i_{f} |
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\psi_{0}=-x_{0} i_{0} |
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\psi_{f}=-x_{m d} i_{d}+\left(x_{f}+x_{c}\right) i_{f}+\left(x_{m d}+x_{c}\right) i_{D} |
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\psi_{D}=-x_{m d} i_{d}+\left(x_{D}+x_{c}\right) i_{D}+\left(x_{m d}+x_{c}\right) i_{f} |
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\psi_{Q 1}=-x_{m q} i_{q}+x_{Q 1} i_{Q 1}+x_{m q } i_{Q 2} |
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\psi_{Q 2}=-x_{m q} i_{q}+x_{Q 2} i_{Q 2}+x_{m q} i_{Q 1} |
Where,
Xc = Canay Reactance (pu)
Wr = Electrical speed (Rad/s)
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x_{d}=x_{m d}+x_{l} |
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x_{q}=x_{m q}+x_{l} |
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x_{f}=x_{m d}+x_{l f} |
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x_{D}=x_{m d}+x_{l D} |
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x_{Q 1}=x_{m q}+x_{l Q 1} |
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x_{Q 2}=x_{m q}+x_{l Q 2} |
The electromagnetic torque (p.u.) developed by the synchronous machine is given by:
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t_{e m}=\psi_{d} i_{q}-\psi_{q} i_{d} |
The torque (p.u.) developed by the excitor is given by:
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t_{e x c}=\frac{v_{f} i_{f}}{\varpi_{e x c}} |
Where Wexc is the mechanical speed (pu) of the exciter
The mechanical (general) equation of mass i is described as follows:
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\begin{array}{l}
{2 H_{i} \frac{d \omega_{i m}}{d t}=T_{i m}-T_{e m}-T_{e x c}-K_{d i} \omega_{i m}-D_{i}\left(\omega_{i m}-\omega_{s m}\right)-D_{i, i-1}\left(\omega_{i m}-\omega_{(i-1) m}\right)-D_{i, i+1}\left(\omega_{i m}-\omega_{(i+1) m}\right)}{-K_{i, i-1}\left(\delta_{i m}-\delta_{(i-1) m}\right)-K_{i, i+1}\left(\delta_{i m}-\delta_{(i+1) m}\right)}
\end{array} |
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\frac{d \delta_{\text {im }}}{d t}=\omega_{\text {im}}-\omega_{s m} |
Where,
: The mechanical torque developed on the ith mass (Nm)Mathinline body --uriencoded--T_%7Bim%7D
: The electromagnetic torque of the synchronous machine (Nm)Mathinline body --uriencoded--T_%7Bem%7D
: The torque developed by the exciter (Nm)Mathinline body --uriencoded--T_%7Bexc%7D
: The mechanical speed of mass i (rad / s)Mathinline body --uriencoded--\omega_%7Bim%7D
: Mechanical synchronous speed (rad / s)Mathinline body --uriencoded--\omega_%7Bsm%7D
: The mechanical angle of mass i with respect to a frame of referenceMathinline body --uriencoded--\delta_%7Bim%7D
: Absolute damping coefficient (Nm / mechanical rad / s)Mathinline body --uriencoded--K_%7Bdi%7D
: Self damping coefficient (Nm / mechanical rad / s)Mathinline body --uriencoded--D_%7Bi%7D
: Mutual damping coefficient (Nm / mechanical rad / s)Mathinline body --uriencoded--D_%7Bi.j%7D
: Angular stiffness (Nm / mechanical rad)Mathinline body --uriencoded--K_%7Bi,j%7D
: Inertia constant (s)Mathinline body --uriencoded--H_%7Bi%7D
At no load, the rotor voltage (in p.u) seen from the stator is given by,
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E_{f d}=\varpi_{r} L^{uns} _{m d}\frac{v_{f}} {r_{f}} |
Where
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In HYPERSIM, we define the field voltage independent of the variation of the speed (field voltage independent of speed deviation):
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V_{f d}= L^{uns} _{m d}\frac{v_{f}} {r_{f}} |
Thus, the field voltage dependent on the variation of the speed (field voltage dependent on speed deviation) is given by :
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E_{f d}= \varpi_{r}V _{f d} |
At steady state and at the fundamental frequency ωr=1, we have :
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E_{f d}= V _{f d} |
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NOTE: When the machine is used in motor mode, the mechanical torque is negative and positive in generator Tim mode. |
Limitations
This model does not take into account hysteresis. Under certain conditions some models of the machine may differ since the model of the synchronized machine-implemented is non-iterative.
References
[1] Power System Stability and Control, P. Kundur, McGraw-Hill 1994
[2] G. Sybille, Tarik Zabaiou, Emergency Diesel-Generator and Asynchronous Motor, Mathworks demo example, version: 2017B
[3] Power System Control and Stability, 2nd edition, P. Mr. Anderson, A.A.Fouad, IEEE press 2003