This page documents the mathematical model of the synchronous machine implemented in RAMSES. The model uses the Equal-Mutual-Flux-Linkage (EMFL) per unit system and supports detailed (round rotor, salient pole) and simplified (no damper) configurations through model switches.
To accommodate different rotor configurations within a single model, integer “model switches” are defined:
Switch Meaning S d 1 S_{d1} S d 1 1 if there is a damper winding d 1 d1 d 1 S q 1 S_{q1} S q 1 1 if there is a damper winding q 1 q1 q 1 S q 2 S_{q2} S q 2 1 if there is an equivalent winding q 2 q2 q 2
Model Switches Detailed, round rotor S d 1 = 1 , S q 1 = 1 , S q 2 = 1 S_{d1} = 1,\; S_{q1} = 1,\; S_{q2} = 1 S d 1 = 1 , S q 1 = 1 , S q 2 = 1 Detailed, salient pole S d 1 = 1 , S q 1 = 1 , S q 2 = 0 S_{d1} = 1,\; S_{q1} = 1,\; S_{q2} = 0 S d 1 = 1 , S q 1 = 1 , S q 2 = 0 Simplified, no damper S d 1 = 0 , S q 1 = 0 , S q 2 = 0 S_{d1} = 0,\; S_{q1} = 0,\; S_{q2} = 0 S d 1 = 0 , S q 1 = 0 , S q 2 = 0
Using the EMFL per unit system, the relationship between magnetic flux linkages and currents is:
( ψ d ψ f ψ d 1 ) = ( L ℓ + M d M d S d 1 M d M d L ℓ f + M d S d 1 M d S d 1 M d S d 1 M d L ℓ d 1 + S d 1 M d ) ( i d i f i d 1 ) \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} ψ d ψ f ψ d 1 = L ℓ + M d M d S d 1 M d M d L ℓ f + M d S d 1 M d S d 1 M d S d 1 M d L ℓ d 1 + S d 1 M d i d i f i d 1 ( ψ q ψ q 1 ψ q 2 ) = ( L ℓ + M q S q 1 M q S q 2 M q S q 1 M q L ℓ q 1 + S q 1 M q S q 2 M q S q 2 M q S q 2 M q L ℓ q 2 + S q 2 M q ) ( i q i q 1 i q 2 ) \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} ψ q ψ q 1 ψ q 2 = L ℓ + M q S q 1 M q S q 2 M q S q 1 M q L ℓ q 1 + S q 1 M q S q 2 M q S q 2 M q S q 2 M q L ℓ q 2 + S q 2 M q i q i q 1 i q 2 The d d d q q q
ψ a d = M d ( i d + i f + S d 1 i d 1 ) \psi_{ad} = M_d(i_d + i_f + S_{d1} i_{d1}) ψ a d = M d ( i d + i f + S d 1 i d 1 ) ψ a q = M q ( i q + S q 1 i q 1 + S q 2 i q 2 ) \psi_{aq} = M_q(i_q + S_{q1} i_{q1} + S_{q2} i_{q2}) ψ a q = M q ( i q + S q 1 i q 1 + S q 2 i q 2 ) Individual flux linkages in terms of air-gap flux:
ψ d = L ℓ i d + ψ a d \psi_d = L_\ell i_d + \psi_{ad} ψ d = L ℓ i d + ψ a d ψ f = L ℓ f i f + ψ a d \psi_f = L_{\ell f} i_f + \psi_{ad} ψ f = L ℓ f i f + ψ a d ψ d 1 = L ℓ d 1 i d 1 + S d 1 ψ a d \psi_{d1} = L_{\ell d1} i_{d1} + S_{d1} \psi_{ad} ψ d 1 = L ℓ d 1 i d 1 + S d 1 ψ a d ψ q = L ℓ i q + ψ a q \psi_q = L_\ell i_q + \psi_{aq} ψ q = L ℓ i q + ψ a q ψ q 1 = L ℓ q 1 i q 1 + S q 1 ψ a q \psi_{q1} = L_{\ell q1} i_{q1} + S_{q1} \psi_{aq} ψ q 1 = L ℓ q 1 i q 1 + S q 1 ψ a q ψ q 2 = L ℓ q 2 i q 2 + S q 2 ψ a q \psi_{q2} = L_{\ell q2} i_{q2} + S_{q2} \psi_{aq} ψ q 2 = L ℓ q 2 i q 2 + S q 2 ψ a q Rotor currents from flux linkages:
i f = ψ f − ψ a d L ℓ f , i d 1 = ψ d 1 − S d 1 ψ a d L ℓ d 1 , i q 1 = ψ q 1 − S q 1 ψ a q L ℓ q 1 , i q 2 = ψ q 2 − S q 2 ψ a q L ℓ q 2 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}} i f = L ℓ f ψ f − ψ a d , i d 1 = L ℓ d 1 ψ d 1 − S d 1 ψ a d , i q 1 = L ℓ q 1 ψ q 1 − S q 1 ψ a q , i q 2 = L ℓ q 2 ψ q 2 − S q 2 ψ a q Let M d u M_d^u M d u M q u M_q^u M q u M d M_d M d M q M_q M q
M d = M d u 1 + m ( ψ a d 2 + ψ a q 2 ) n M_d = \frac{M_d^u}{1 + m \left(\sqrt{\psi_{ad}^2 + \psi_{aq}^2}\right)^n} M d = 1 + m ( ψ a d 2 + ψ a q 2 ) n M d u M q = M q u 1 + m ( ψ a d 2 + ψ a q 2 ) n M_q = \frac{M_q^u}{1 + m \left(\sqrt{\psi_{ad}^2 + \psi_{aq}^2}\right)^n} M q = 1 + m ( ψ a d 2 + ψ a q 2 ) n M q u where m m m n n n SYNC_MACH record.
Substituting into the air-gap flux expressions yields the algebraic equations:
ψ a d ( 1 + m ( ψ a d 2 + ψ a q 2 ) n M d u + 1 L ℓ f + S d 1 L ℓ d 1 ) − i d − 1 L ℓ f ψ f − S d 1 L ℓ d 1 ψ d 1 = 0 \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 ψ a d M d u 1 + m ( ψ a d 2 + ψ a q 2 ) n + L ℓ f 1 + L ℓ d 1 S d 1 − i d − L ℓ f 1 ψ f − L ℓ d 1 S d 1 ψ d 1 = 0 ψ a q ( 1 + m ( ψ a d 2 + ψ a q 2 ) n M q u + S q 1 L ℓ q 1 + S q 2 L ℓ q 2 ) − i q − S q 1 L ℓ q 1 ψ q 1 − S q 2 L ℓ q 2 ψ q 2 = 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 ψ a q M q u 1 + m ( ψ a d 2 + ψ a q 2 ) n + L ℓ q 1 S q 1 + L ℓ q 2 S q 2 − i q − L ℓ q 1 S q 1 ψ q 1 − L ℓ q 2 S q 2 ψ q 2 = 0 The d d d q q q ( x , y ) (x, y) ( x , y ) δ \delta δ
( v d v q ) = ( − sin δ cos δ cos δ sin δ ) ( v x v y ) \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} ( v d v q ) = ( − sin δ cos δ cos δ sin δ ) ( v x v y ) ( i d i q ) = ( − sin δ cos δ cos δ sin δ ) ( i x i y ) \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} ( i d i q ) = ( − sin δ cos δ cos δ sin δ ) ( i x i y ) After transformation, the air-gap flux algebraic equations in ( x , y ) (x, y) ( x , y )
ψ a d ( 1 + m ( ψ a d 2 + ψ a q 2 ) n M d u + 1 L ℓ f + S d 1 L ℓ d 1 ) + sin δ i x − cos δ i y − 1 L ℓ f ψ f − S d 1 L ℓ d 1 ψ d 1 = 0 \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 ψ a d M d u 1 + m ( ψ a d 2 + ψ a q 2 ) n + L ℓ f 1 + L ℓ d 1 S d 1 + sin δ i x − cos δ i y − L ℓ f 1 ψ f − L ℓ d 1 S d 1 ψ d 1 = 0 ψ a q ( 1 + m ( ψ a d 2 + ψ a q 2 ) n M q u + S q 1 L ℓ q 1 + S q 2 L ℓ q 2 ) − cos δ i x − sin δ i y − S q 1 L ℓ q 1 ψ q 1 − S q 2 L ℓ q 2 ψ q 2 = 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 ψ a q M q u 1 + m ( ψ a d 2 + ψ a q 2 ) n + L ℓ q 1 S q 1 + L ℓ q 2 S q 2 − cos δ i x − sin δ i y − L ℓ q 1 S q 1 ψ q 1 − L ℓ q 2 S q 2 ψ q 2 = 0 0 = sin δ v x − cos δ v y + ( R a sin δ − ω L ℓ cos δ ) i x − ( R a cos δ + ω L ℓ sin δ ) i y − ω ψ a q 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 = sin δ v x − cos δ v y + ( R a sin δ − ω L ℓ cos δ ) i x − ( R a cos δ + ω L ℓ sin δ ) i y − ω ψ a q 0 = − cos δ v x − sin δ v y − ( R a cos δ + ω N L ℓ sin δ ) i x − ( R a sin δ − ω L ℓ cos δ ) i y + ω ψ a d 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} 0 = − cos δ v x − sin δ v y − ( R a cos δ + ω N L ℓ sin δ ) i x − ( R a sin δ − ω L ℓ cos δ ) i y + ω ψ a d d ψ f d t = ω N ( K f v f − R f ψ f − ψ a d L ℓ f ) \frac{d\psi_f}{dt} = \omega_N \left( K_f v_f - R_f \frac{\psi_f - \psi_{ad}}{L_{\ell f}} \right) d t d ψ f = ω N ( K f v f − R f L ℓ f ψ f − ψ a d ) d ψ d 1 d t = − ω N R d 1 ψ d 1 − S d 1 ψ a d L ℓ d 1 \frac{d\psi_{d1}}{dt} = -\omega_N R_{d1} \frac{\psi_{d1} - S_{d1} \psi_{ad}}{L_{\ell d1}} d t d ψ d 1 = − ω N R d 1 L ℓ d 1 ψ d 1 − S d 1 ψ a d d ψ q 1 d t = − ω N R q 1 ψ q 1 − S q 1 ψ a q L ℓ q 1 \frac{d\psi_{q1}}{dt} = -\omega_N R_{q1} \frac{\psi_{q1} - S_{q1} \psi_{aq}}{L_{\ell q1}} d t d ψ q 1 = − ω N R q 1 L ℓ q 1 ψ q 1 − S q 1 ψ a q d ψ q 2 d t = − ω N R q 2 ψ q 2 − S q 2 ψ a q L ℓ q 2 \frac{d\psi_{q2}}{dt} = -\omega_N R_{q2} \frac{\psi_{q2} - S_{q2} \psi_{aq}}{L_{\ell q2}} d t d ψ q 2 = − ω N R q 2 L ℓ q 2 ψ q 2 − S q 2 ψ a q 1 ω N d δ d t = ω − ω c o i \frac{1}{\omega_N} \frac{d\delta}{dt} = \omega - \omega_{coi} ω N 1 d t d δ = ω − ω co i 2 H d ω d t = K m T m − T e − D ( ω − ω c o i ) 2H \frac{d\omega}{dt} = K_m T_m - T_e - D(\omega - \omega_{coi}) 2 H d t d ω = K m T m − T e − D ( ω − ω co i ) where the electromagnetic torque T e T_e T e
T e = ψ a d i q − ψ a q i d = ψ a d ( cos δ i x + sin δ i y ) − ψ a q ( − sin δ i x + cos δ i y ) 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) T e = ψ a d i q − ψ a q i d = ψ a d ( cos δ i x + sin δ i y ) − ψ a q ( − sin δ i x + cos δ i y ) The model has 10 state variables : i x i_x i x i y i_y i y ψ a d \psi_{ad} ψ a d ψ a q \psi_{aq} ψ a q ψ f \psi_f ψ f ψ d 1 \psi_{d1} ψ d 1 ψ q 1 \psi_{q1} ψ q 1 ψ q 2 \psi_{q2} ψ q 2 δ \delta δ ω \omega ω
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
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:
I B R A T I O = I f B m a c I f B e x c IBRATIO = \frac{I_{fB}^{mac}}{I_{fB}^{exc}} I B R A T I O = I f B e x c I f B ma c where I f B m a c I_{fB}^{mac} I f B ma c I f B e x c I_{fB}^{exc} I f B e x c
I B R A T I O = i f , p u e x c i f , p u m a c IBRATIO = \frac{i_{f,pu}^{exc}}{i_{f,pu}^{mac}} I B R A T I O = i f , p u ma c i f , p u e x c Open-circuit unsaturated machine (most common): I f B e x c I_{fB}^{exc} I f B e x c V = 1 V = 1 V = 1 ω = 1 \omega = 1 ω = 1
I B R A T I O = M d u = X d u − X ℓ IBRATIO = M_d^u = X_d^u - X_\ell I B R A T I O = M d u = X d u − X ℓ Open-circuit saturated machine : Same conditions but with saturation:
I B R A T I O = M d u 1 + m = X d u − X ℓ 1 + m IBRATIO = \frac{M_d^u}{1 + m} = \frac{X_d^u - X_\ell}{1 + m} I B R A T I O = 1 + m M d u = 1 + m X d u − X ℓ Saturated machine at nominal operating conditions : I f B e x c I_{fB}^{exc} I f B e x c P = cos ϕ N P = \cos\phi_N P = cos ϕ N Q = sin ϕ N Q = \sin\phi_N Q = sin ϕ N
The synchronous machine is declared in the data file with the SYNC_MACH record:
SYNC_MACH name bus FP FQ P Q SNOM Pnom H D IBRATIO
XT/RL Xl Xd X'd X"d Xq X'q X"q m n Ra T'do T"do T'qo T"qo
EXC exc_type param1 param2 ...
TOR tor_type param1 param2 ... ;
Parameter Description Unit nameMachine name (max 8 characters) — busConnection bus name (max 8 characters) — FPActive power participation fraction (0–1) — FQReactive power participation fraction (0–1) — PInitial active power (used when FP = 0) MW QInitial reactive power (used when FQ = 0) Mvar SNOMNominal apparent power MVA PnomNominal active power of the turbine MW HInertia constant s DDamping coefficient pu IBRATIOField current base ratio I f B m a c / I f B e x c I_{fB}^{mac}/I_{fB}^{exc} I f B ma c / I f B e x c pu XT/RLKeyword: XT for step-up transformer reactance, RL for line resistance — Value after XT/RL Step-up transformer reactance or line resistance pu XlLeakage reactance L ℓ L_\ell L ℓ pu Xdd-axis synchronous reactance pu X'dd-axis transient reactance pu X"dd-axis subtransient reactance pu Xqq-axis synchronous reactance pu X'qq-axis transient reactance (use * to set equal to X'd) pu X"qq-axis subtransient reactance (use * to set equal to X"d) pu mSaturation coefficient (use * for default) — nSaturation exponent (use * for default) — RaArmature resistance pu T'dod-axis open-circuit transient time constant s T"dod-axis open-circuit subtransient time constant s T'qoq-axis open-circuit transient time constant (use * for round-rotor default) s T"qoq-axis open-circuit subtransient time constant s
All reactances and resistances are in per unit on the machine base (S n o m S_{nom} S n o m
The EXC and TOR sub-records specify the excitation system and turbine-governor models. See the Model Reference for available models.
The FP, FQ, P, Q fields control how the machine’s initial operating point is determined from the power flow solution. See Reference Frames & Initialization for details.
