This page describes the most common synchronous machine models used in stability studies.
Nomenclature
The standard machine parameters are defined as follows:
is the armature resistance (pu)
is the armature reactance (pu)
is the d-axis synchronous reactance (pu)
is the q-axis synchronous reactance (pu)
is the d-axis transient reactance (pu)
is the q-axis transient reactance (pu)
is the d-axis subtransient reactance (pu)
is the q-axis subtransient reactance (pu)
is the d-axis transient open loop time constant (s)
is the q-axis transient open loop time constant (s)
is the d-axis subtransient open loop time constant (s)
is the q-axis subtransient open loop time constant (s)
is the machine inertia constant (MWs/MVA)
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:
Sauer, P.W., Pai, M. A., "Power System Dynamics and Stability", Stipes Publishing, 2006
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]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a467558da38e529780457833f1e425c31643f0f9)
![{\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]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a59eedb6be32df1a4f4ab87feea9cf7eecb7dc)
![{\displaystyle {\dot {\psi ''_{d}}}={\frac {1}{T''_{d0}}}\left[E'_{q}-\psi ''_{d}-(X'_{d}-X_{a})I_{d}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9784fba490cb2eb973b502af946b2200d8e0ec8)
![{\displaystyle {\dot {\psi ''_{q}}}={\frac {1}{T''_{q0}}}\left[-E'_{d}-\psi ''_{q}-(X'_{q}-X_{a})I_{q}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2535664a1b692013006074c748f39f4b35007950)


where



Stator electrical equations (neglecting electromagnetic transients):


Equations of motion:
![{\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77e21374261e9d47a84ac6485dd65a498a0cb957)

Initialisation:














6th Order (Anderson-Fouad) Model
6th order synchronous machine model based on the book:
Anderson, P. M., Fouad, A. A., "Power System Control and Stability", Wiley-IEEE Press, New York, 2002
Stator magnetic equations:
![{\displaystyle {\dot {E'_{q}}}={\frac {1}{T'_{d0}}}\left[V_{fd}-(X_{d}-X'_{d})I_{d}-E'_{q}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/def5964f2e752544829546a6a7ff3327b35b2652)
![{\displaystyle {\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[(X_{q}-X'_{q})I_{q}-E'_{d}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f202e3cfb9a2156ce5bd3393b35a853019e8554)
![{\displaystyle {\dot {E''_{q}}}={\frac {1}{T''_{d0}}}\left[E'_{q}-(X'_{d}-X''_{d})-E''_{q}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cbfe00d1afa3f57864d5c73774edc727ac730f2)
![{\displaystyle {\dot {E''_{d}}}={\frac {1}{T''_{q0}}}\left[E'_{d}-(X'_{q}-X''_{q})-E''_{d}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ee6063b834d8a0b1e3ff1a2dc5eee2a2934646)




Stator electrical equations (neglecting electromagnetic transients):


Equations of motion:
![{\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77e21374261e9d47a84ac6485dd65a498a0cb957)

Initialisation:














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]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/def5964f2e752544829546a6a7ff3327b35b2652)
![{\displaystyle {\dot {E'_{d}}}={\frac {1}{T'_{q0}}}\left[(X_{q}-X'_{q})I_{q}-E'_{d}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f202e3cfb9a2156ce5bd3393b35a853019e8554)




Stator electrical equations (neglecting electromagnetic transients):


Equations of motion:
![{\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77e21374261e9d47a84ac6485dd65a498a0cb957)

Initialisation:













2nd Order (Classical) Model
Stator equations:


Equations of motion:
![{\displaystyle {\dot {\omega }}={\frac {1}{2H}}\left[P_{m}-P_{e}-D(\omega -\omega _{s})\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77e21374261e9d47a84ac6485dd65a498a0cb957)

Initialisation:




