# Synchronous Machine Models

This page describes the most common synchronous machine models used in stability studies.

## Nomenclature

The standard machine parameters are defined as follows:

• ${\displaystyle R_{a}\,}$ is the armature resistance (pu)
• ${\displaystyle X_{a}\,}$ is the armature reactance (pu)
• ${\displaystyle X_{d}\,}$ is the d-axis synchronous reactance (pu)
• ${\displaystyle X_{q}\,}$ is the q-axis synchronous reactance (pu)
• ${\displaystyle X'_{d}\,}$ is the d-axis transient reactance (pu)
• ${\displaystyle X'_{q}\,}$ is the q-axis transient reactance (pu)
• ${\displaystyle X''_{d}\,}$ is the d-axis subtransient reactance (pu)
• ${\displaystyle X''_{q}\,}$ is the q-axis subtransient reactance (pu)
• ${\displaystyle T'_{d0}\,}$ is the d-axis transient open loop time constant (s)
• ${\displaystyle T'_{q0}\,}$ is the q-axis transient open loop time constant (s)
• ${\displaystyle T''_{d0}\,}$ is the d-axis subtransient open loop time constant (s)
• ${\displaystyle T''_{q0}\,}$ is the q-axis subtransient open loop time constant (s)
• ${\displaystyle H\,}$ is the machine inertia constant (MWs/MVA)
• ${\displaystyle D\,}$ is an additional damping constant (pu)

Note that per-unit values are usually expressed on the machine's MVA base.

## 6th Order (Sauer-Pai) Model

6th order synchronous machine model based on the book:

Stator magnetic equations:

${\displaystyle {\dot {E'_{q}}}={\frac {1}{T'_{d0}}}\left[V_{fd}-E'_{q}-(X_{d}-X'_{d})\left(I_{d}-\gamma _{d2}\psi ''_{d}-(1-\gamma _{d1})I_{d}+\gamma _{d2}E'_{q}\right)\right]\,}$
${\displaystyle {\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[-E'_{q}-(X_{q}-X'_{q})\left(I_{q}-\gamma _{q2}\psi ''_{q}-(1-\gamma _{q1})I_{q}-\gamma _{q2}E'_{d}\right)\right]\,}$
${\displaystyle {\dot {\psi ''_{d}}}={\frac {1}{T''_{d0}}}\left[E'_{q}-\psi ''_{d}-(X'_{d}-X_{a})I_{d}\right]\,}$
${\displaystyle {\dot {\psi ''_{q}}}={\frac {1}{T''_{q0}}}\left[-E'_{d}-\psi ''_{q}-(X'_{q}-X_{a})I_{q}\right]\,}$
${\displaystyle \psi _{d}=-X''_{d}I_{d}+\gamma _{d1}E'_{q}+(1-\gamma _{d1})\psi ''_{d}\,}$
${\displaystyle \psi _{q}=-X''_{q}I_{q}-\gamma _{q1}E'_{d}+(1-\gamma _{q1})\psi ''_{q}\,}$

where ${\displaystyle \gamma _{d1}={\frac {X''_{d}-X_{a}}{X'_{d}-X_{a}}}\,}$

${\displaystyle \gamma _{q1}={\frac {X''_{q}-X_{a}}{X'_{q}-X_{a}}}\,}$
${\displaystyle \gamma _{d2}={\frac {1-\gamma _{d1}}{X'_{d}-X_{a}}}\,}$
${\displaystyle \gamma _{q2}={\frac {1-\gamma _{q1}}{X'_{q}-X_{a}}}\,}$

Stator electrical equations (neglecting electromagnetic transients):

${\displaystyle V_{d}=-\omega \psi _{q}-R_{a}I_{d}\,}$
${\displaystyle V_{q}=\omega \psi _{d}-R_{a}I_{q}\,}$

Equations of motion:

${\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}$
${\displaystyle {\dot {\delta }}=\Omega _{s}(\omega -\omega _{s})\,}$

Initialisation:

${\displaystyle {\boldsymbol {E}}_{q0}={\boldsymbol {V}}_{t0}+(R_{a}+jX_{q})\times {\boldsymbol {I}}_{a0}\,}$
${\displaystyle \delta _{0}=\angle {\boldsymbol {E}}_{q0}\,}$
${\displaystyle \psi _{0}=\angle {\boldsymbol {I}}_{a0}\,}$
${\displaystyle I_{d0}=|{\boldsymbol {I}}_{a0}|\sin(\delta _{0}-\psi _{0})\,}$
${\displaystyle I_{q0}=|{\boldsymbol {I}}_{a0}|\cos(\delta _{0}-\psi _{0})\,}$
${\displaystyle V_{d0}=|{\boldsymbol {V}}_{t0}|\sin(\delta _{0}-\theta _{0})\,}$
${\displaystyle V_{q0}=|{\boldsymbol {V}}_{t0}|\cos(\delta _{0}-\theta _{0})\,}$
${\displaystyle E'_{d0}=V_{d}-X''_{q}I_{q0}+R_{a}I_{d0}-(1-\gamma _{q1})(X'_{q}-X_{a})I_{q0}\,}$
${\displaystyle E'_{q0}=V_{q}+X''_{d}I_{d0}+R_{a}I_{q0}+(1-\gamma _{d1})(X'_{d}-X_{a})I_{d0}\,}$
${\displaystyle \psi ''_{d0}=E'_{q0}-(X'_{d}-X_{a})I_{d0}\,}$
${\displaystyle \psi ''_{q0}=-E'_{d0}-(X'_{q}-X_{a})I_{q0}\,}$
${\displaystyle V_{fd0}=E'_{q0}+(X_{d}-X'_{d})\left(I_{d0}-\gamma _{d2}\psi ''_{d0}-(1-\gamma _{d1})I_{d0}+\gamma _{d2}E'_{q0}\right)\,}$
${\displaystyle P_{m}=P_{e0}=(V_{d0}+R_{a}I_{d0})I_{d0}+(V_{q0}+R_{a}I_{q0})I_{q0}\,}$
${\displaystyle \omega _{0}=\omega _{s}\,}$

6th order synchronous machine model based on the book:

Stator magnetic equations:

${\displaystyle {\dot {E'_{q}}}={\frac {1}{T'_{d0}}}\left[V_{fd}-(X_{d}-X'_{d})I_{d}-E'_{q}\right]\,}$
${\displaystyle {\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[(X_{q}-X'_{q})I_{q}-E'_{d}\right]\,}$
${\displaystyle {\dot {E''_{q}}}={\frac {1}{T''_{d0}}}\left[E'_{q}-(X'_{d}-X''_{d})-E''_{q}\right]\,}$
${\displaystyle {\dot {E''_{d}}}={\frac {1}{T''_{q0}}}\left[E'_{d}-(X'_{q}-X''_{q})-E''_{d}\right]\,}$
${\displaystyle E''_{q}-V_{q}=R_{a}I_{q}+X''_{d}I_{d}\,}$
${\displaystyle E''_{d}-V_{d}=R_{a}I_{d}-X''_{q}I_{q}\,}$
${\displaystyle \psi _{d}=E''_{q}-X''_{d}I_{d}\,}$
${\displaystyle \psi _{q}=-E''_{d}-X''_{q}I_{q}\,}$

Stator electrical equations (neglecting electromagnetic transients):

${\displaystyle V_{d}=-\omega \psi _{q}-R_{a}I_{d}\,}$
${\displaystyle V_{q}=\omega \psi _{d}-R_{a}I_{q}\,}$

Equations of motion:

${\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}$
${\displaystyle {\dot {\delta }}=\Omega _{s}(\omega -\omega _{s})\,}$

Initialisation:

${\displaystyle {\boldsymbol {E}}_{q0}={\boldsymbol {V}}_{t0}+(R_{a}+jX_{q})\times {\boldsymbol {I}}_{a0}\,}$
${\displaystyle \delta _{0}=\angle {\boldsymbol {E}}_{q0}\,}$
${\displaystyle \phi _{0}=\angle {\boldsymbol {I}}_{a0}\,}$
${\displaystyle I_{d0}=|{\boldsymbol {I}}_{a0}|\sin(\delta _{0}-\phi _{0})\,}$
${\displaystyle I_{q0}=|{\boldsymbol {I}}_{a0}|\cos(\delta _{0}-\phi _{0})\,}$
${\displaystyle V_{fd0}=|{\boldsymbol {E}}_{q0}|+(X_{d}-X_{q})I_{d0}\,}$
${\displaystyle E'_{q0}=V_{fd0}-(X_{d}-X'_{d})I_{d0}\,}$
${\displaystyle E''_{q0}=E'_{q0}-(X'_{d}-X''_{d})I_{d0}\,}$
${\displaystyle E'_{d0}=(X_{q}-X'_{q})I_{q0}\,}$
${\displaystyle E''_{d0}=E'_{d0}+(X'_{q}-X''_{q})I_{q0}\,}$
${\displaystyle V_{d0}=E''_{d0}+X''_{q}I_{q0}-R_{a}I_{d0}\,}$
${\displaystyle V_{q0}=E''_{q0}-X''_{d}I_{d0}-R_{a}I_{q0}\,}$
${\displaystyle P_{m}=P_{e0}=(V_{d0}+R_{a}I_{d0})I_{d0}+(V_{q0}+R_{a}I_{q0})I_{q0}\,}$
${\displaystyle \omega _{0}=\omega _{s}\,}$

## 4th Order (Two-Axis) Model

Stator magnetic equations:

${\displaystyle {\dot {E'_{q}}}={\frac {1}{T'_{d0}}}\left[V_{fd}-(X_{d}-X'_{d})I_{d}-E'_{q}\right]\,}$
${\displaystyle {\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[(X_{q}-X'_{q})I_{q}-E'_{d}\right]\,}$
${\displaystyle E'_{q}-V_{q}=R_{a}I_{q}+X'_{d}I_{d}\,}$
${\displaystyle E'_{d}-V_{d}=R_{a}I_{d}-X'_{q}I_{q}\,}$
${\displaystyle \psi _{d}=E'_{q}-X'_{d}I_{d}\,}$
${\displaystyle \psi _{q}=-E'_{d}-X'_{q}I_{q}\,}$

Stator electrical equations (neglecting electromagnetic transients):

${\displaystyle V_{d}=-\omega \psi _{q}-R_{a}I_{d}\,}$
${\displaystyle V_{q}=\omega \psi _{d}-R_{a}I_{q}\,}$

Equations of motion:

${\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}$
${\displaystyle {\dot {\delta }}=\Omega _{s}(\omega -\omega _{s})\,}$

Initialisation:

${\displaystyle {\boldsymbol {E}}_{q0}={\boldsymbol {V}}_{t0}+(R_{a}+jX_{q})\times {\boldsymbol {I}}_{a0}\,}$
${\displaystyle \delta _{0}=\angle {\boldsymbol {E}}_{q0}\,}$
${\displaystyle \phi _{0}=\angle {\boldsymbol {I}}_{a0}\,}$
${\displaystyle \theta _{0}=\angle {\boldsymbol {V}}_{t0}\,}$
${\displaystyle I_{d0}=|{\boldsymbol {I}}_{a0}|\sin(\delta _{0}-\phi _{0})\,}$
${\displaystyle I_{q0}=|{\boldsymbol {I}}_{a0}|\cos(\delta _{0}-\phi _{0})\,}$
${\displaystyle V_{d0}=|{\boldsymbol {V}}_{t0}|\sin(\delta _{0}-\theta _{0})\,}$
${\displaystyle V_{q0}=|{\boldsymbol {V}}_{t0}|\cos(\delta _{0}-\theta _{0})\,}$
${\displaystyle E'_{q0}=V_{q0}+R_{a}I_{q0}+X'_{d}I_{d0}\,}$
${\displaystyle E'_{d0}=V_{d0}+R_{a}I_{d0}-X'_{q}I_{q0}\,}$
${\displaystyle V_{fd0}=E'_{q0}+(X_{d}-X'_{d})I_{d0}\,}$
${\displaystyle P_{m}=P_{e0}=(V_{d0}+R_{a}I_{d0})I_{d0}+(V_{q0}+R_{a}I_{q0})I_{q0}\,}$
${\displaystyle \omega _{0}=\omega _{s}\,}$

## 2nd Order (Classical) Model

Stator equations:

${\displaystyle E'_{q}-V_{q}=R_{a}I_{q}+X'_{d}I_{d}\,}$
${\displaystyle V_{d}=X'_{d}I_{q}-R_{a}I_{d}\,}$

Equations of motion:

${\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}$
${\displaystyle {\dot {\delta }}=\Omega _{s}(\omega -\omega _{s})\,}$

Initialisation:

${\displaystyle {\boldsymbol {E}}_{q0}={\boldsymbol {V}}_{t0}+(R_{a}+jX'_{d})\times {\boldsymbol {I}}_{a0}\,}$
${\displaystyle \delta _{0}=\angle {\boldsymbol {E}}_{q0}\,}$
${\displaystyle \theta _{0}=\angle {\boldsymbol {V}}_{t0}\,}$
${\displaystyle P_{m}=P_{e0}=\left({\frac {1}{R_{a}+jX'_{d}}}\right)|{\boldsymbol {V}}_{t0}||{\boldsymbol {E}}_{q0}|\sin(\delta _{0}-\theta _{0})\,}$
${\displaystyle \omega _{0}=\omega _{s}\,}$