# Synchronous Machine Models

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

## Nomenclature

The standard machine parameters are defined as follows:

• $R_a \,$ is the armature resistance (pu)
• $X_a \,$ is the armature reactance (pu)
• $X_d \,$ is the d-axis synchronous reactance (pu)
• $X_q \,$ is the q-axis synchronous reactance (pu)
• $X'_{d} \,$ is the d-axis transient reactance (pu)
• $X'_{q} \,$ is the q-axis transient reactance (pu)
• $X''_{d} \,$ is the d-axis subtransient reactance (pu)
• $X''_{q} \,$ is the q-axis subtransient reactance (pu)
• $T'_{d0} \,$ is the d-axis transient open loop time constant (s)
• $T'_{q0} \,$ is the q-axis transient open loop time constant (s)
• $T''_{d0} \,$ is the d-axis subtransient open loop time constant (s)
• $T''_{q0} \,$ is the q-axis subtransient open loop time constant (s)
• $H \,$ is the machine inertia constant (MWs/MVA)
• $D \,$ 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:

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

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

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

Stator electrical equations (neglecting electromagnetic transients):

$V_{d} = -\omega \psi_{q} - R_{a} I_{d} \,$
$V_{q} = \omega \psi_{d} - R_{a} I_{q} \,$

Equations of motion:

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

Initialisation:

$\boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X_q) \times \boldsymbol{I}_{a0} \,$
$\delta_0 = \angle \boldsymbol{E}_{q0} \,$
$\psi_0 = \angle \boldsymbol{I}_{a0} \,$
$I_{d0} = |\boldsymbol{I}_{a0}| \sin (\delta_0 - \psi_0 ) \,$
$I_{q0} = |\boldsymbol{I}_{a0}| \cos (\delta_0 - \psi_0 ) \,$
$V_{d0} = |\boldsymbol{V}_{t0}| \sin (\delta_0 - \theta_0 ) \,$
$V_{q0} = |\boldsymbol{V}_{t0}| \cos (\delta_0 - \theta_0 ) \,$
$E'_{d0} = V_{d} - X''_{q} I_{q0} + R_{a} I_{d0} - (1 - \gamma_{q1}) (X'_{q} - X_{a}) I_{q0} \,$
$E'_{q0} = V_{q} + X''_{d} I_{d0} + R_{a} I_{q0} + (1 - \gamma_{d1}) (X'_{d} - X_{a}) I_{d0} \,$
$\psi''_{d0} = E'_{q0} - (X'_{d} - X_{a}) I_{d0} \,$
$\psi''_{q0} = -E'_{d0} - (X'_{q} - X_{a}) I_{q0} \,$
$V_{fd0} = E'_{q0} + (X_{d} - X'_{d}) \left( I_{d0} - \gamma_{d2} \psi''_{d0} - (1 - \gamma_{d1}) I_{d0} + \gamma_{d2} E'_{q0} \right) \,$
$P_m = P_{e0} = (V_{d0} + R_{a} I_{d0}) I_{d0} + (V_{q0} + R_{a} I_{q0}) I_{q0} \,$
$\omega_0 = \omega_s \,$

6th order synchronous machine model based on the book:

Stator magnetic equations:

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

Stator electrical equations (neglecting electromagnetic transients):

$V_{d} = -\omega \psi_{q} - R_{a} I_{d} \,$
$V_{q} = \omega \psi_{d} - R_{a} I_{q} \,$

Equations of motion:

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

Initialisation:

$\boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X_q) \times \boldsymbol{I}_{a0} \,$
$\delta_0 = \angle \boldsymbol{E}_{q0} \,$
$\phi_0 = \angle \boldsymbol{I}_{a0} \,$
$I_{d0} = |\boldsymbol{I}_{a0}| \sin (\delta_0 - \phi_0 ) \,$
$I_{q0} = |\boldsymbol{I}_{a0}| \cos (\delta_0 - \phi_0 ) \,$
$V_{fd0} = |\boldsymbol{E}_{q0}| + (X_{d} - X_{q}) I_{d0} \,$
$E'_{q0} = V_{fd0} - (X_{d} - X'_{d}) I_{d0} \,$
$E''_{q0} = E'_{q0} - (X'_{d} - X''_{d}) I_{d0} \,$
$E'_{d0} = (X_{q} - X'_{q}) I_{q0} \,$
$E''_{d0} = E'_{d0} + (X'_{q} - X''_{q}) I_{q0} \,$
$V_{d0} = E''_{d0} + X''_{q} I_{q0} - R_{a} I_{d0} \,$
$V_{q0} = E''_{q0} - X''_{d} I_{d0} - R_{a} I_{q0} \,$
$P_m = P_{e0} = (V_{d0} + R_{a} I_{d0}) I_{d0} + (V_{q0} + R_{a} I_{q0}) I_{q0} \,$
$\omega_0 = \omega_s \,$

## 4th Order (Two-Axis) Model

Stator magnetic equations:

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

Stator electrical equations (neglecting electromagnetic transients):

$V_{d} = -\omega \psi_{q} - R_{a} I_{d} \,$
$V_{q} = \omega \psi_{d} - R_{a} I_{q} \,$

Equations of motion:

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

Initialisation:

$\boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X_q) \times \boldsymbol{I}_{a0} \,$
$\delta_0 = \angle \boldsymbol{E}_{q0} \,$
$\phi_0 = \angle \boldsymbol{I}_{a0} \,$
$\theta_0 = \angle \boldsymbol{V}_{t0} \,$
$I_{d0} = |\boldsymbol{I}_{a0}| \sin (\delta_0 - \phi_0 ) \,$
$I_{q0} = |\boldsymbol{I}_{a0}| \cos (\delta_0 - \phi_0 ) \,$
$V_{d0} = |\boldsymbol{V}_{t0}| \sin (\delta_0 - \theta_0 ) \,$
$V_{q0} = |\boldsymbol{V}_{t0}| \cos (\delta_0 - \theta_0 ) \,$
$E'_{q0} = V_{q0} + R_a I_{q0} + X'_{d} I_{d0} \,$
$E'_{d0} = V_{d0} + R_a I_{d0} - X'_{q} I_{q0} \,$
$V_{fd0} = E'_{q0} + (X_d - X'_d) I_{d0} \,$
$P_m = P_{e0} = (V_{d0} + R_{a} I_{d0}) I_{d0} + (V_{q0} + R_{a} I_{q0}) I_{q0} \,$
$\omega_0 = \omega_s \,$

## 2nd Order (Classical) Model

Stator equations:

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

Equations of motion:

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

Initialisation:

$\boldsymbol{E}_{q0} = \boldsymbol{V}_{t0} + (R_a + j X'_d) \times \boldsymbol{I}_{a0} \,$
$\delta_0 = \angle \boldsymbol{E}_{q0} \,$
$\theta_0 = \angle \boldsymbol{V}_{t0} \,$
$P_m = P_{e0} = \left( \frac{1}{R_a + j X'_d} \right) |\boldsymbol{V}_{t0}| |\boldsymbol{E}_{q0}| \sin(\delta_0 - \theta_0) \,$
$\omega_0 = \omega_s \,$