Salient-pole synchronous machine with fundamental parameterization

Machines / Synchronous Machine (Salient Pole)

The Synchronous Machine Salient Pole (fundamental) block models a salient-pole synchronous machine using fundamental parameters.

The synchronous machine equations are expressed with respect to a rotating reference frame defined by the equation

$${\theta}_{e}(t)=N{\theta}_{r}(t),$$

where:

*θ*is the electrical angle._{e}*N*is the number of pole pairs.*θ*is the rotor angle._{r}

Park’s transformation maps the synchronous machine equations to the rotating reference frame with respect to the electrical angle. Park’s transformation is defined by

$${P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta}_{e}& \mathrm{cos}({\theta}_{e}-\frac{2\pi}{3})& \mathrm{cos}({\theta}_{e}+\frac{2\pi}{3})\\ -\mathrm{sin}{\theta}_{e}& -\mathrm{sin}({\theta}_{e}-\frac{2\pi}{3})& -\mathrm{sin}({\theta}_{e}+\frac{2\pi}{3})\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$$

Park’s transformation is used to define the per-unit synchronous machine equations. The stator voltage equations are defined by

$${e}_{d}=\frac{1}{{\omega}_{base}}\frac{\text{d}{\psi}_{d}}{\text{d}t}-{\Psi}_{q}{\omega}_{r}-{R}_{a}{i}_{d},$$

${e}_{q}=\frac{1}{{\omega}_{base}}\frac{\text{d}{\psi}_{q}}{\text{d}t}+{\Psi}_{d}{\omega}_{r}-{R}_{a}{i}_{q},$

and

$${e}_{0}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{0}}{dt}-{R}_{a}{i}_{0},$$

where:

*e*,_{d}*e*, and_{q}*e*are the_{0}*d*-axis,*q*-axis, and zero-sequence stator voltages, defined by$$\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\\ {e}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$$

*v*,_{a}*v*, and_{b}*v*are the stator voltages measured from port ~ to neutral port n._{c}*ω*is the per-unit base electrical speed._{base}*ψ*,_{d}*ψ*, and_{q}*ψ*are the_{0}*d*-axis,*q*-axis, and zero-sequence stator flux linkages.*ω*is the per-unit rotor rotational speed._{r}*R*is the stator resistance._{a}*i*,_{d}*i*, and_{q}*i*are the_{0}*d*-axis,*q*-axis, and zero-sequence stator currents, defined by$$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$$

*i*,_{a}*i*, and_{b}*i*are the stator currents flowing from port ~ to port n._{c}

The rotor voltage equations are defined by

$${e}_{fd}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{fd}}{dt}+{R}_{fd}{i}_{fd},$$

$${e}_{1d}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{1d}}{dt}+{R}_{1d}{i}_{1d}=0,$$

and

$${e}_{1}{}_{q}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{1q}}{dt}+{R}_{1q}{i}_{1q}=0,$$

where:

*e*is the field voltage._{fd}*e*, and_{1d}*e*are the voltages across the_{1q}*d*-axis damper winding 1 and*q*-axis damper winding 1. They are equal to 0.*ψ*,_{fd}*ψ*, and_{1d}*ψ*are the magnetic fluxes linking the field circuit,_{1q}*d*-axis damper winding 1, and*q*-axis damper winding 1.*R*,_{fd}*R*, and_{1d}*R*are the resistances of rotor field circuit,_{1q}*d*-axis damper winding 1, and*q*-axis damper winding 1.*i*,_{fd}*i*, and_{1d}*i*are the currents flowing in the field circuit,_{1q}*d*-axis damper winding 1, and*q*-axis damper winding 1.

The saturation equations are defined by

$${\psi}_{ad}={\psi}_{d}+{L}_{l}{i}_{d},$$

$${\psi}_{aq}={\psi}_{q}+{L}_{l}{i}_{q},$$

$${\psi}_{at}=\sqrt{{\psi}_{ad}^{2}+{\psi}_{aq}^{2}},$$

$${K}_{s}=1$$ (If saturation is disabled),

${K}_{s}=f\left({\psi}_{at}\right)$ (If saturation is enabled),

and

${L}_{ad}={K}_{s}*{L}_{adu},$

where:

*ψ*is the_{ad}*d*-axis air-gap or mutual flux linkage.*ψ*is the_{aq}*q*-axis air-gap or mutual flux linkage.*ψ*is the air-gap flux linkage._{at}*K*is the saturation factor._{s}*L*is the unsaturated mutual inductance of the stator_{adu}*d*-axis.*L*is the mutual inductance of the stator_{ad}*d*-axis.

The saturation factor function, f, is calculated from the per-unit open-circuit lookup table as:

${L}_{ad}=\frac{d{\psi}_{at}}{d{i}_{fd}},$

${V}_{ag}=g({i}_{fd}),$

and

${L}_{ad}=\frac{dg({i}_{fd})}{d{i}_{fd}}=\frac{d{V}_{ag}}{d{i}_{fd}},$

where:

*V*is the per-unit air-gap voltage._{ag}

In per-unit,

${K}_{s}=\frac{{L}_{ad}}{{L}_{adu}},$

and

${\psi}_{at}={V}_{ag}$

can be rearranged to

${K}_{s}=f({\psi}_{at}).$

The stator flux linkage equations are defined by

$${\Psi}_{d}=-({L}_{ad}+{L}_{l}){i}_{d}\text{}+{L}_{ad}{i}_{fd}+{L}_{ad}{i}_{1d},$$

$$\Psi q=-({L}_{aq}+{L}_{l}){i}_{q}\text{}+{L}_{aq}{i}_{1q},$$

and

$${\Psi}_{0}=-{L}_{0}{i}_{0},$$

where:

*L*is the stator leakage inductance._{l}*L*and_{ad}*L*are the mutual inductances of the stator_{aq}*d*-axis and*q*-axis.

The rotor flux linkage equations are defined by

$${\psi}_{fd}={L}_{ffd}{i}_{fd}+{L}_{f1d}{i}_{1d}-{L}_{ad}{i}_{d},$$

$${\psi}_{1d}={L}_{f1d}{i}_{fd}+{L}_{11d}{i}_{1d}-{L}_{ad}{i}_{d},$$

and

$${\psi}_{1q}={L}_{11q}{i}_{1q}-{L}_{aq}{i}_{q},$$

where:

*L*,_{ffd}*L*, and_{11d}*L*are the self-inductances of the rotor field circuit,_{11q}*d*-axis damper winding 1, and*q*-axis damper winding 1.*L*is the rotor field circuit and_{f1d}*d*-axis damper winding 1 mutual inductance. They are defined by the following equations.$${L}_{ffd}={L}_{ad}+{L}_{fd}$$

$${L}_{f1d}={L}_{ffd}-{L}_{fd}$$

$${L}_{11d}={L}_{f1d}+{L}_{1d}$$

$${L}_{11q}={L}_{aq}+{L}_{1q}$$

These equations assume that per-unit mutual inductance *L _{12q}* =

The rotor torque is defined by

$${T}_{e}={\Psi}_{d}{i}_{q}-{\Psi}_{q}{i}_{d}.$$

You can perform plotting and display actions using the **Power
Systems** menu on the block context menu.

Right-click the block and, from the **Power
Systems** menu, select an option:

**Display Base Values**displays the machine per-unit base values in the MATLAB^{®}Command Window.**Display Associated Base Values**displays associated per-unit base values in the MATLAB Command Window.**Display Associated Initial Conditions**displays associated initial conditions in the MATLAB Command Window.**Plot Open-Circuit Saturation (pu)**plots air-gap voltage,*V*, versus field current,_{ag}*i*, both measured in per-unit, in a MATLAB figure window. The plot contains three traces:_{fd}Unsaturated:

**Stator d-axis mutual inductance (unsaturated), Ladu**you specifySaturated:

**Per-unit open-circuit lookup table (Vag versus ifd)**you specifyDerived: Open-circuit lookup table (per-unit) derived from the

**Per-unit open-circuit lookup table (Vag versus ifd)**you specify. This data is used to calculate the saturation factor,*K*, versus magnetic flux linkage,_{s}*ψ*, characteristic._{at}

**Plot Saturation Factor (pu)**plots saturation factor,*K*, versus magnetic flux linkage,_{s}*ψ*, both measured in per-unit, in a MATLAB figure window using the present machine parameters. This parameter is derived from other parameters that you specify:_{at}**Stator d-axis mutual inductance (unsaturated), Ladu****Per-unit field current saturation data, ifd****Per-unit air-gap voltage saturation data, Vag**

**Rated apparent power**Rated apparent power. The default value is

`300e6`

`V*A`

.**Rated voltage**RMS rated line-line voltage. The default value is

`24e3`

`V`

.**Rated electrical frequency**Nominal electrical frequency at which rated apparent power is quoted. The default value is

`60`

`Hz`

.**Number of pole pairs**Number of machine pole pairs. The default value is

`10`

.**Specify field circuit input required to produce rated terminal voltage at no load by**Choose between

`Field circuit voltage`

and`Field circuit current`

. The default value is`Field circuit current`

.**Field circuit current**This parameter is visible only when

**Specify field circuit input required to produce rated terminal voltage at no load by**is set to`Field circuit current`

. The default value is`1000`

`A`

.**Field circuit voltage**This parameter is visible only when

**Specify field circuit input required to produce rated terminal voltage at no load by**is set to`Field circuit voltage`

. The default value is`216.54`

V.

**Stator d-axis mutual inductance (unsaturated), Ladu**Unsaturated stator

*d*-axis mutual inductance,*L*. If_{adu}**Magnetic saturation representation**is set to`None`

, this is equivalent to the stator*d*-axis mutual inductance,*L*. The default value is_{ad}`0.9`

pu.**Stator q-axis mutual inductance, Laq**Stator

*q*-axis mutual inductance, Laq. The default value is`0.55`

pu.**Stator zero-sequence inductance, L0**Stator zero-sequence inductance, L0. The default value is

`0.15`

pu.**Stator leakage inductance, Ll**Stator leakage inductance. The default value is

`0.15`

pu.**Stator resistance, Ra**Stator resistance. The default value is

`0.011`

pu.**Rotor field circuit inductance, Lfd**Rotor field circuit inductance. The default value is

`0.2571`

pu.**Rotor field circuit resistance, Rfd**Rotor field circuit resistance. The default value is

`0.0006`

pu.**Rotor d-axis damper winding 1 inductance, L1d**Rotor

*d*-axis damper winding 1 inductance. The default value is`0.2`

pu.**Rotor d-axis damper winding 1 resistance, R1d**Rotor

*d*-axis damper winding 1 resistance. The default value is`0.0354`

pu.**Rotor q-axis damper winding 1 inductance, L1q**Rotor

*q*-axis damper winding 1 inductance. The default value is`0.2567`

pu.**Rotor q-axis damper winding 1 resistance, R1q**Rotor

*q*-axis damper winding 1 resistance. The default value is`0.0428`

pu.

**Magnetic saturation representation**Block magnetic saturation representation. Options are:

`None`

`Per-unit open-circuit lookup table (Vag versus ifd)`

The default value is

`None`

.**Per-unit field current saturation data, ifd**The field current,

*i*, data populates the air-gap voltage,_{fd}*V*, versus field current,_{ag}*i*, lookup table. This parameter is only visible when you set_{fd}**Magnetic saturation representation**to`Per-unit open-circuit lookup table (Vag versus ifd)`

. This parameter must contain a vector with at least five elements. The default value is`[0.00, 0.48, 0.76, 1.38, 1.79]`

pu.**Per-unit air-gap voltage saturation data, Vag**The air-gap voltage,

*V*, data populates the air-gap voltage,_{ag}*V*, versus field current,_{ag}*i*, lookup table. This parameter is only visible when you set_{fd}**Magnetic saturation representation**to`Per-unit open-circuit lookup table (Vag versus ifd)`

. This parameter must contain a vector with at least five elements. The default value is`[0.00 0.43 0.59 0.71 0.76]`

pu.

**Specify initialization by**Select between

`Electrical power and voltage output`

and`Mechanical and magnetic states`

. The default value is`Electrical power and voltage output`

.**Terminal voltage magnitude**Initial RMS line-line voltage. This parameter is visible only when you set

**Specify initialization by**to`Electrical power and voltage output`

. The default value is`24e3`

`V`

.**Terminal voltage angle**Initial voltage angle. This parameter is visible only when you set

**Specify initialization by**to`Electrical power and voltage output`

. The default value is`0`

`deg`

.**Terminal active power**Initial active power. This parameter is visible only when

**Specify initialization by**is set to`Electrical power and voltage output`

. The default value is`270e6`

`V*A`

.**Terminal reactive power**Initial reactive power. This parameter is visible only when you set

**Specify initialization by**to`Electrical power and voltage output`

. The default value is`0`

`V*A`

.**Initial rotor angle**Initial rotor angle. During steady-state operation, set this parameter to the sum of the load angle and required terminal voltage offset. This parameter is visible only when you set

**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

`deg`

.**Initial stator d-axis magnetic flux linkage**Stator

*d*-axis initial flux linkage. This parameter is visible only when you set**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.**Initial stator q-axis magnetic flux linkage**Stator

*q*-axis initial flux linkage. This parameter is visible only when you set**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.**Initial stator zero-sequence magnetic flux linkage**Zero-sequence initial flux linkage. This parameter is visible only when you set

**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.**Initial field circuit magnetic flux linkage**Field circuit initial flux linkage. This parameter is visible only when you set

**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.**Initial d-axis damper winding 1 magnetic flux linkage**The

*d*-axis damper winding 1 initial flux linkage. This parameter is visible only when you set**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.**Initial q-axis damper winding 1 magnetic flux linkage**The

*q*-axis damper winding 1 initial flux linkage. This parameter is visible only when you set**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.**Initial q-axis damper winding 2 magnetic flux linkage**The

*q*-axis damper winding 2 initial flux linkage. This parameter is visible only when you set**Specify initialization by**to`Mechanical and magnetic states`

. The default value is`0`

pu.

The block has the following ports:

`fd+`

Electrical conserving port corresponding to the field winding positive terminal

`fd-`

Electrical conserving port corresponding to the field winding negative terminal

`R`

Mechanical rotational conserving port associated with the machine rotor

`C`

Mechanical rotational conserving port associated with the machine case

`pu`

Physical signal vector port associated with the machine per-unit measurements. The vector elements are:

`pu_fd_Efd`

`pu_fd_Ifd`

`pu_torque`

`pu_velocity`

`pu_ed`

`pu_eq`

`pu_e0`

`pu_id`

`pu_iq`

`pu_i0`

`~`

Expandable three-phase port associated with the stator windings

`n`

Electrical conserving port associated with the neutral point of the wye winding configuration

[1] Kundur, P. *Power System Stability and Control.* New
York, NY: McGraw Hill, 1993.

[2] Lyshevski, S. E. *Electromechanical Systems,
Electric Machines and Applied Mechatronics.* Boca Raton,
FL: CRC Press, 1999.

Synchronous Machine Measurement | Synchronous Machine Model 2.1 (fundamental) | Synchronous Machine Model 2.1 (standard) | Synchronous Machine Round Rotor (fundamental) | Synchronous Machine Round Rotor (standard) | Synchronous Machine Salient Pole (standard)

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