Synchronous Machine Model

This page documents the mathematical model of the synchronous machine implemented in RAMSES. The model is a detailed sixth-order model, including four rotor windings with saturation effects. It uses the Equal-Mutual-Flux-Linkage (EMFL) per unit system and supports detailed (round rotor, salient-pole) and simplified (field winding only) configurations through model switches.

Model Switches

To accommodate different rotor configurations within a single model, integer “model switches” are defined:

Switch	Meaning
$S_{d1}$	1 if there is a damper winding $d1$ , 0 otherwise
$S_{q1}$	1 if there is a damper winding $q1$ , 0 otherwise
$S_{q2}$	1 if there is an equivalent winding $q2$ , 0 otherwise

Model	Switches
Detailed, round rotor	$S_{d1} = 1,\; S_{q1} = 1,\; S_{q2} = 1$
Detailed, salient-pole rotor	$S_{d1} = 1,\; S_{q1} = 1,\; S_{q2} = 0$
Detailed, salient-pole rotor	$S_{d1} = 1,\; S_{q1} = 0,\; S_{q2} = 1$
Simplified, field winding only	$S_{d1} = 0,\; S_{q1} = 0,\; S_{q2} = 0$

The second and third combinations yield the same results. Models with fewer rotor windings are specified by skipping the corresponding data in the SYNC_MACH record (see below).

Park Transformation

The well-known Park transformation is used to replace time-varying inductances and oscillatory stator currents and voltages with constant values. The machine is represented by equivalent windings along the direct ( $d$ ) and quadrature ( $q$ ) axes — a field winding $f$ and damper winding $d1$ on the $d$ axis, and windings $q1$ , $q2$ on the $q$ axis:

Synchronous machine windings Equivalent windings of the Park transformation

Flux-Current Relationships

Using the EMFL per unit system, the relationship between magnetic flux linkages and currents is:

\begin{pmatrix} \psi_d \\ \psi_f \\ \psi_{d1} \end{pmatrix} = \begin{pmatrix} L_\ell + M_d & M_d & S_{d1} M_d \\ M_d & L_{\ell f} + M_d & S_{d1} M_d \\ S_{d1} M_d & S_{d1} M_d & L_{\ell d1} + S_{d1} M_d \end{pmatrix} \begin{pmatrix} i_d \\ i_f \\ i_{d1} \end{pmatrix}

\begin{pmatrix} \psi_q \\ \psi_{q1} \\ \psi_{q2} \end{pmatrix} = \begin{pmatrix} L_\ell + M_q & S_{q1} M_q & S_{q2} M_q \\ S_{q1} M_q & L_{\ell q1} + S_{q1} M_q & S_{q2} M_q \\ S_{q2} M_q & S_{q2} M_q & L_{\ell q2} + S_{q2} M_q \end{pmatrix} \begin{pmatrix} i_q \\ i_{q1} \\ i_{q2} \end{pmatrix}

The $d$ and $q$ components of the air-gap flux are:

\psi_{ad} = M_d(i_d + i_f + S_{d1} i_{d1})

\psi_{aq} = M_q(i_q + S_{q1} i_{q1} + S_{q2} i_{q2})

Individual flux linkages in terms of air-gap flux:

\psi_d = L_\ell i_d + \psi_{ad}

\psi_f = L_{\ell f} i_f + \psi_{ad}

\psi_{d1} = L_{\ell d1} i_{d1} + S_{d1} \psi_{ad}

\psi_q = L_\ell i_q + \psi_{aq}

\psi_{q1} = L_{\ell q1} i_{q1} + S_{q1} \psi_{aq}

\psi_{q2} = L_{\ell q2} i_{q2} + S_{q2} \psi_{aq}

Rotor currents from flux linkages:

i_f = \frac{\psi_f - \psi_{ad}}{L_{\ell f}}, \qquad i_{d1} = \frac{\psi_{d1} - S_{d1} \psi_{ad}}{L_{\ell d1}}, \qquad i_{q1} = \frac{\psi_{q1} - S_{q1} \psi_{aq}}{L_{\ell q1}}, \qquad i_{q2} = \frac{\psi_{q2} - S_{q2} \psi_{aq}}{L_{\ell q2}}

Saturation Model

Let $M_d^u$ and $M_q^u$ be the unsaturated direct- and quadrature-axis mutual inductances. The saturated values $M_d$ and $M_q$ are:

M_d = \frac{M_d^u}{1 + m \left(\sqrt{\psi_{ad}^2 + \psi_{aq}^2}\right)^n}

M_q = \frac{M_q^u}{1 + m \left(\sqrt{\psi_{ad}^2 + \psi_{aq}^2}\right)^n}

where $m$ and $n$ are the saturation exponents specified in the SYNC_MACH record.

Substituting into the air-gap flux expressions yields the algebraic equations:

\psi_{ad} \left( \frac{1 + m(\sqrt{\psi_{ad}^2 + \psi_{aq}^2})^n}{M_d^u} + \frac{1}{L_{\ell f}} + \frac{S_{d1}}{L_{\ell d1}} \right) - i_d - \frac{1}{L_{\ell f}} \psi_f - \frac{S_{d1}}{L_{\ell d1}} \psi_{d1} = 0

\psi_{aq} \left( \frac{1 + m(\sqrt{\psi_{ad}^2 + \psi_{aq}^2})^n}{M_q^u} + \frac{S_{q1}}{L_{\ell q1}} + \frac{S_{q2}}{L_{\ell q2}} \right) - i_q - \frac{S_{q1}}{L_{\ell q1}} \psi_{q1} - \frac{S_{q2}}{L_{\ell q2}} \psi_{q2} = 0

Reference Frame

All synchronous machines have their rotor positions referred to the $x$ axis of the network reference frame (see Reference Frames & Initialization). The rotor angle $\delta$ of a machine is the angle difference between its $q$ axis and the $x$ reference axis. In steady state, the machine internal emf (proportional to field current) is aligned along the $q$ axis; $\delta$ is thus the phase angle of that emf with respect to the $x$ axis.

Definition of the rotor angle delta

The $d$ and $q$ components of the stator voltage and current relate to the network $(x, y)$ components through the rotor angle $\delta$ :

\begin{pmatrix} v_d \\ v_q \end{pmatrix} = \begin{pmatrix} -\sin\delta & \cos\delta \\ \cos\delta & \sin\delta \end{pmatrix} \begin{pmatrix} v_x \\ v_y \end{pmatrix}

\begin{pmatrix} i_d \\ i_q \end{pmatrix} = \begin{pmatrix} -\sin\delta & \cos\delta \\ \cos\delta & \sin\delta \end{pmatrix} \begin{pmatrix} i_x \\ i_y \end{pmatrix}

After transformation, the air-gap flux algebraic equations in $(x, y)$ coordinates become:

\psi_{ad} \left( \frac{1 + m(\sqrt{\psi_{ad}^2 + \psi_{aq}^2})^n}{M_d^u} + \frac{1}{L_{\ell f}} + \frac{S_{d1}}{L_{\ell d1}} \right) + \sin\delta\, i_x - \cos\delta\, i_y - \frac{1}{L_{\ell f}} \psi_f - \frac{S_{d1}}{L_{\ell d1}} \psi_{d1} = 0

\psi_{aq} \left( \frac{1 + m(\sqrt{\psi_{ad}^2 + \psi_{aq}^2})^n}{M_q^u} + \frac{S_{q1}}{L_{\ell q1}} + \frac{S_{q2}}{L_{\ell q2}} \right) - \cos\delta\, i_x - \sin\delta\, i_y - \frac{S_{q1}}{L_{\ell q1}} \psi_{q1} - \frac{S_{q2}}{L_{\ell q2}} \psi_{q2} = 0

Park Equations

Original form

v_d = -R_a i_d - \omega \psi_q \qquad\qquad v_q = -R_a i_q + \omega \psi_d

\frac{d\psi_f}{dt} = \omega_N (K_f v_f - R_f i_f) \qquad \frac{d\psi_{d1}}{dt} = -\omega_N R_{d1} i_{d1} \qquad \frac{d\psi_{q1}}{dt} = -\omega_N R_{q1} i_{q1} \qquad \frac{d\psi_{q2}}{dt} = -\omega_N R_{q2} i_{q2}

where $R_a$ is the stator (armature) resistance, $R_f$ the field winding resistance, $R_{d1}$ , $R_{q1}$ , $R_{q2}$ the rotor winding resistances, $\omega$ the rotor speed (pu), $\omega_N = 2\pi f_{nom}$ the nominal angular frequency (rad/s), $v_f$ the field voltage, and $K_f$ a coefficient to pass from per unit values of the excitation system to per unit values of the machine.

The stator equations are transformed to the $(x, y)$ frame using the rotation matrices above, and the rotor currents are eliminated using the flux-current relationships, yielding the equations actually solved by RAMSES:

Stator equations (algebraic, in $x$ - $y$ frame)

0 = \sin\delta\, v_x - \cos\delta\, v_y + (R_a \sin\delta - \omega L_\ell \cos\delta)\, i_x - (R_a \cos\delta + \omega L_\ell \sin\delta)\, i_y - \omega \psi_{aq}

0 = -\cos\delta\, v_x - \sin\delta\, v_y - (R_a \cos\delta + \omega_N L_\ell \sin\delta)\, i_x - (R_a \sin\delta - \omega L_\ell \cos\delta)\, i_y + \omega \psi_{ad}

Rotor equations (differential)

\frac{d\psi_f}{dt} = \omega_N \left( K_f v_f - R_f \frac{\psi_f - \psi_{ad}}{L_{\ell f}} \right)

\frac{d\psi_{d1}}{dt} = -\omega_N R_{d1} \frac{\psi_{d1} - S_{d1} \psi_{ad}}{L_{\ell d1}}

\frac{d\psi_{q1}}{dt} = -\omega_N R_{q1} \frac{\psi_{q1} - S_{q1} \psi_{aq}}{L_{\ell q1}}

\frac{d\psi_{q2}}{dt} = -\omega_N R_{q2} \frac{\psi_{q2} - S_{q2} \psi_{aq}}{L_{\ell q2}}

Rotor Motion

\frac{1}{\omega_N} \frac{d\delta}{dt} = \omega - \omega_{ref}

2H \frac{d\omega}{dt} = K_m T_m - T_e - D(\omega - \omega_{ref})

where $H$ is the inertia constant (in s), $T_m$ the mechanical torque produced by the turbine, $K_m$ a coefficient to pass from per unit values of the turbine to per unit values of the machine, and $\omega_{ref}$ the angular speed of the reference axes — $\omega_{coi}$ in the COI reference frame, or 1 pu in the synchronous frame (selected by the $OMEGA_REF solver setting).

The electromagnetic torque $T_e$ is:

T_e = \psi_{ad} i_q - \psi_{aq} i_d = \psi_{ad}(\cos\delta\, i_x + \sin\delta\, i_y) - \psi_{aq}(-\sin\delta\, i_x + \cos\delta\, i_y)

State Variables and Equations Summary

The model has 10 state variables: $i_x$ , $i_y$ , $\psi_{ad}$ , $\psi_{aq}$ , $\psi_f$ , $\psi_{d1}$ , $\psi_{q1}$ , $\psi_{q2}$ , $\delta$ , $\omega$ .

These are balanced by:

4 algebraic equations: air-gap flux (d and q), stator voltage (d and q)
6 differential equations: field flux, d1 damper flux, q1 damper flux, q2 damper flux, rotor angle, rotor speed

Per Unit System and IBRATIO

The synchronous machine model uses the EMFL per unit system, while the excitation system typically uses its own per unit system. The parameter IBRATIO bridges these two bases:

IBRATIO = \frac{I_{fB}^{mac}}{I_{fB}^{exc}}

where $I_{fB}^{mac}$ is the field winding base current in the machine model and $I_{fB}^{exc}$ is the base current in the excitation system model. The relationship between per-unit field currents in the two systems is:

IBRATIO = \frac{i_{f,pu}^{exc}}{i_{f,pu}^{mac}}

Common per unit conventions for IBRATIO

Open-circuit unsaturated machine (most common): $I_{fB}^{exc}$ is the field current that produces nominal stator voltage ( $V = 1$ pu) at nominal speed ( $\omega = 1$ pu) with the stator open, neglecting saturation:

IBRATIO = M_d^u = X_d^u - X_\ell

Open-circuit saturated machine: Same conditions but with saturation:

IBRATIO = \frac{M_d^u}{1 + m} = \frac{X_d^u - X_\ell}{1 + m}

Saturated machine at nominal operating conditions: $I_{fB}^{exc}$ is the field current when the machine produces nominal active and reactive powers ( $P = \cos\phi_N$ , $Q = \sin\phi_N$ ) at nominal voltage and speed, with saturation.

SYNC_MACH Record

The machine model requires the nominal system frequency, given by the mandatory FNOM record (see Solver Settings):

FNOM F ;

where F is the nominal frequency in Hz.

The synchronous machine itself is declared with the SYNC_MACH record:

SYNC_MACH name bus FP FQ P Q SNOM Pnom H D IBRATIO
          TYPE_MOD  <14 machine parameters, see below>
          EXC exc_type param1 param2 ...
          TOR tor_type param1 param2 ... ;

TYPE_MOD is a keyword selecting which of two equivalent parameter formats the 14 machine parameters that follow are given in:

RL — the inductances and resistances of the Park model are supplied directly:
```
RL  Ll Mdu Llf Lld1 Mqu Llq1 Llq2 m n Ra Rf Rd1 Rq1 Rq2
```
XT — characteristic reactances and open-circuit time constants are supplied; RAMSES converts them internally to the Park parameters (see Parameter Conversion):
```
XT  Xl Xd X'd X"d Xq X'q X"q m n Ra T'do T"do T'qo T"qo
```

Common parameters

Parameter	Description	Unit
`name`	Machine name (max 20 characters)
`bus`	Connection bus name (max 8 characters)
`FP`	Active power participation fraction (0–1)
`FQ`	Reactive power participation fraction (0–1)
`P`	Initial active power (used when FP = 0)	MW
`Q`	Initial reactive power (used when FQ = 0)	Mvar
`SNOM`	Nominal apparent power, used as base power in the machine model	MVA
`Pnom`	Nominal active power of the turbine, used as base power for the turbine model	MW
`H`	Inertia constant	s
`D`	Damping coefficient (usually set to zero when the damper windings are modelled)	pu
`IBRATIO`	Field current base ratio $I_{fB}^{mac}/I_{fB}^{exc}$ (see above)	pu
`TYPE_MOD`	Parameter format keyword: `RL` or `XT` (case-insensitive)

Machine parameters — `XT` format

Parameter	Description	Unit
`Xl`	Leakage reactance $L_\ell$	pu
`Xd`	d-axis synchronous reactance ( $M_d^u = X_d - X_\ell$ )	pu
`X'd`	d-axis transient reactance (must be smaller than `Xd`)	pu
`X"d`	d-axis subtransient reactance (`*` if no $d1$ damper winding)	pu
`Xq`	q-axis synchronous reactance ( $M_q^u = X_q - X_\ell$ )	pu
`X'q`	q-axis transient reactance (`*` if no $q1$ winding; must be smaller than `Xq`)	pu
`X"q`	q-axis subtransient reactance (`*` if no $q2$ winding)	pu
`m`	Saturation coefficient (set to 0 to neglect saturation)
`n`	Saturation exponent (ignored when `m` = 0)
`Ra`	Armature resistance	pu
`T'do`	d-axis open-circuit transient time constant	s
`T"do`	d-axis open-circuit subtransient time constant (`*` if no $d1$ damper winding)	s
`T'qo`	q-axis open-circuit transient time constant (`*` if no $q1$ winding)	s
`T"qo`	q-axis open-circuit subtransient time constant (`*` if no $q2$ winding)	s

Machine parameters — `RL` format

Parameter	Description	Unit
`Ll`	Stator leakage inductance $L_\ell$	pu
`Mdu`	Unsaturated d-axis mutual inductance $M_d^u$	pu
`Llf`	Field winding leakage inductance $L_{\ell f}$	pu
`Lld1`	$d1$ damper leakage inductance $L_{\ell d1}$ (`*` if no $d1$ winding)	pu
`Mqu`	Unsaturated q-axis mutual inductance $M_q^u$	pu
`Llq1`	$q1$ winding leakage inductance $L_{\ell q1}$ (`*` if no $q1$ winding)	pu
`Llq2`	$q2$ winding leakage inductance $L_{\ell q2}$ (`*` if no $q2$ winding)	pu
`m`	Saturation coefficient (set to 0 to neglect saturation)
`n`	Saturation exponent (ignored when `m` = 0)
`Ra`	Armature resistance	pu
`Rf`	Field winding resistance $R_f$	pu
`Rd1`	$d1$ damper resistance $R_{d1}$ (`*` if no $d1$ winding)	pu
`Rq1`	$q1$ winding resistance $R_{q1}$ (`*` if no $q1$ winding)	pu
`Rq2`	$q2$ winding resistance $R_{q2}$ (`*` if no $q2$ winding)	pu

Omitting rotor circuits

A rotor circuit the machine does not have is skipped by putting * in both of its fields — the inductance/reactance and the matching resistance/time-constant field (specifying only one is an error):

Circuit	`RL` fields	`XT` fields	Model switch set to 0
$d1$ damper	`Lld1`, `Rd1`	`X"d`, `T"do`	$S_{d1}$
$q1$ winding	`Llq1`, `Rq1`	`X'q`, `T'qo`	$S_{q1}$
$q2$ winding	`Llq2`, `Rq2`	`X"q`, `T"qo`	$S_{q2}$

The combination $S_{d1} = 0$ with $S_{q1} = S_{q2} = 1$ (field winding plus both q-axis windings but no d-axis damper) is rejected. In the XT format, if the fitted Park parameters come out negative, RAMSES logs an “unrealistic Park inductances or resistances” warning — the supplied reactances and time constants are physically inconsistent.

All reactances, inductances and resistances are in per unit on the machine base ( $S_{nom}$ , nominal voltage), using the EMFL per unit system for the rotor quantities. Time constants are entered in seconds and normalised internally by $t_b = 1/(2\pi f_{nom})$ .

The EXC and TOR sub-records specify the excitation system and turbine-governor models. See the Model Reference for available models.

Initialization Output

At initialization RAMSES prints one block per synchronous machine. Example:

NUMBER OF SYNCHRONOUS MACHINES :    1

machine              at bus                 V            P           Q        delta    sat    island  br
                     excit model          vf(pu)   torque model               Tm(pu)

G5                   5                    1.0000    450.00186     68.49769    70.99   1.0000       1    1
                     exc_GENERIC3         2.3680   THERMAL_GENERIC1          0.97826

Here the machine G5, connected to bus 5, is in service (br = 1) and injects about 450 MW and 68 Mvar into the grid under a bus voltage of 1 pu.

delta is the initial value of the rotor angle $\delta$ , in degrees.
sat is the saturation factor $sat = 1 + m \left( \sqrt{\psi_{ad}^2 + \psi_{aq}^2} \right)^n \geq 1$ the ratio between the field current in the saturated machine and the corresponding field current when saturation is neglected, for the same operating conditions. It characterizes the extra excitation current needed in the presence of saturated material ( $sat = 1$ when m is set to zero).
vf is the initial field voltage on the exciter base, which is indirectly defined by the IBRATIO parameter of the machine.
Tm is the initial mechanical torque on the turbine base, which is defined by the Pnom parameter of the machine.

Parameter Conversion (XT ↔ RL)

For a detailed derivation of how STEPSS converts the XT standard parameters (reactances and open-circuit time constants) to the RL Park parameters (inductances and resistances) — including the exact algorithm from the source, known conversion pitfalls when cross-checking against EMT simulators, and a reference Python implementation — see:

➜ Synchronous Machine — Parameter Conversion (XT ↔ RL)