hermess.devices.synchronous

Classes

Synchronous	Metaclass for SG in rectangular coordinates with TGOV1 governor and pluggable AVR
SynchronousTransient	Transient two-axis SG with TGOV1 governor and IEEEDC1A AVR
SynchronousSubtransient	Subtransient Anderson Fouad SG with TGOV1 governor and IEEEDC1A AVR
SynchronousSubtransientSP	Subtransient Sauer and Pai SG model with stator dynamics
SynchronousSubtransientSP6	Subtransient Sauer and Pai SG 6th order model with neglected stator dynamics
SynchronousSubtransientSP6DAE	Subtransient Sauer-Pai 6th-order machine expressed as an explicit DAE.
SynchronousSubtransientSP_DAE	Subtransient Sauer-Pai machine (with stator dynamics) with the stator
GENROU	Sixth-order round-rotor synchronous machine (PSS/E GENROU): two rotor
GENSAL	Fifth-order salient-pole synchronous machine (PSS/E GENSAL): two rotor

Module Contents

class hermess.devices.synchronous.Synchronous(avr: hermess.devices.avr.AVR = None, governor: hermess.devices.governor.Governor = None, pss: hermess.devices.pss.PSS = None, shaft: hermess.devices.shaft.Shaft = None)

Bases: hermess.devices.device.DeviceRect

Metaclass for SG in rectangular coordinates with TGOV1 governor and pluggable AVR

Parameters:

• avr (hermess.devices.avr.AVR)

• governor (hermess.devices.governor.Governor)

• pss (hermess.devices.pss.PSS)

• shaft (hermess.devices.shaft.Shaft)

_shaft: hermess.devices.shaft.Shaft

_avr: hermess.devices.avr.AVR

_governor: hermess.devices.governor.Governor

_pss: hermess.devices.pss.PSS = None

fn

H

R_s

x_d

x_q

x_l

D

f

ns = 0

gcall(dae: hermess.system.Dae, i_d: casadi.SX, i_q: casadi.SX) → None

Parameters:

• dae (hermess.system.Dae)

• i_d (casadi.SX)

• i_q (casadi.SX)

Return type:

None

_omega_ref(dae: hermess.system.Dae) → casadi.SX

Per-machine reference-frame frequency: the nominal dae.omega_net (= 1 p.u.) by default, or the per-bus reference frequency when dae.omega_ref_buses is present.

Parameters:

dae (hermess.system.Dae)

Return type:

casadi.SX

rotor(dae: hermess.system.Dae, Pe: casadi.SX) → None

Delegate the swing (rotor-motion) equations to the pluggable shaft strategy. The default SingleMass writes the classic single-mass swing equation (reading the governor port var_sym("pm"), the air-gap power Pe and the reference frequency); a multi-mass (torsional) shaft adds further rotor masses while keeping the generator mass on the delta / omega names, so the electromagnetic models, the network injection and the controllers are untouched.

Parameters:

• dae (hermess.system.Dae)

• Pe (casadi.SX)

Return type:

None

fgcall(dae: hermess.system.Dae) → None

Template orchestration shared by all machine models: the EM subclass defines stator currents + flux dynamics and returns the air-gap power; the base wires the rotor, the controller ports, and the network.

Parameters:

dae (hermess.system.Dae)

Return type:

None

abstract electromagnetic(dae: hermess.system.Dae)

Subclass hook: define the stator currents (i_d, i_q), write the flux/emf differential equations, and return (i_d, i_q, Pe) where Pe is the air-gap power/torque consumed by rotor().

Parameters:

dae (hermess.system.Dae)

governor_fcall(dae: hermess.system.Dae) → None

Delegate the governor’s equations to the strategy object. The strategy writes its differential equations into dae.f and, if it declares private algebraics (e.g. an algebraic ‘pm’), their residuals into dae.g.

Parameters:

dae (hermess.system.Dae)

Return type:

None

tgov1(dae: hermess.system.Dae) → None

Deprecated: use governor_fcall instead.

Parameters:

dae (hermess.system.Dae)

Return type:

None

avr_fcall(dae: hermess.system.Dae) → None

Delegate the AVR’s equations to the strategy object. The strategy writes its differential equations into dae.f and, if it declares private algebraics (e.g. an algebraic ‘Efd’), their residuals into dae.g.

Parameters:

dae (hermess.system.Dae)

Return type:

None

var_sym(dae: hermess.system.Dae, name: str) → casadi.SX

Resolve a registered variable to its symbol, wherever it lives.

A coupling variable such as ‘Efd’ may be a differential state (in dae.x) or a device-private algebraic (in dae.y), depending on the AVR strategy. This resolver lets the machine’s electromagnetic equations reference it without knowing which, so state-based and algebraic exciters are interchangeable.

Parameters:

• dae (hermess.system.Dae)

• name (str)

Return type:

casadi.SX

pss_fcall(dae: hermess.system.Dae) → None

Delegate the PSS’s equations to the strategy object (no-op if there is no PSS). The strategy writes its differential equations into dae.f and, if it declares private algebraics (e.g. an algebraic ‘Vs’), their residuals into dae.g.

Parameters:

dae (hermess.system.Dae)

Return type:

None

pss_signal(dae: hermess.system.Dae) → casadi.SX

The supplementary stabilizing signal ‘Vs’ summed into the AVR voltage error (the host-mediated PSS->AVR coupling). Returns the ‘Vs’ symbol (state or private algebraic, via var_sym()) when a PSS is present, else 0.

Parameters:

dae (hermess.system.Dae)

Return type:

casadi.SX

ieeedc1a(dae: hermess.system.Dae) → None

Deprecated: use avr_fcall instead.

Parameters:

dae (hermess.system.Dae)

Return type:

None

class hermess.devices.synchronous.SynchronousTransient(avr=None, governor=None, pss=None, shaft=None)

Bases: Synchronous

Transient two-axis SG with TGOV1 governor and IEEEDC1A AVR

Rotor Dynamics

\[\begin{split}\dot{\delta} &= 2 \pi f_n \Delta \omega \\ \Delta \dot{\omega} &= \frac{1}{2 H} \left( P_m - E_d I_d - E_q I_q + (X_q' - X_d') I_d I_q - D \Delta \omega - f (\Delta \omega + 1) \right)\end{split}\]

Electromagnetic Equations

\[\begin{split}\dot{E}_q &= \frac{1}{T_{d'}} \left( -E_q + E_f + (X_d - X_d') I_d \right)\\ \dot{E}_d &= \frac{1}{T_{q'}} \left( -E_d - (X_q - X_q') I_q \right)\end{split}\]

Excitation System Equations

\[\begin{split} \dot{E}_{\textup{fd}} &= \frac{1}{T_{E}} \left( -K_{E,i}E_{\textup{fd},i} + V_{R,i}\right)\\ \dot{R}_{f} &= \frac{1}{T_{F} } \left(- R_{f} + \frac{K_{F}}{T_{F}}E_{\textup{fd}} \right)\\ \dot{V}_{R} &= \frac{1}{T_{A}} \left(-V_{R} + K_{A}R_{f} - \frac{K_{A}K_{F}}{T_{F}}E_{\textup{fd}} + K_{A}(v_{\textup{ref}} - v_i) \right)\end{split}\]

Turbine-Governor System Equations

\[\begin{split}\dot{p}_{\textup{m}} &= \frac{1}{T_{ch}} \left( p_{\textup{sv}}- p_{\textup{m}} \right)\\ \dot{p}_{sv} &= \frac{1}{T_{sv}} \left( - \frac{\Delta \omega}{R_d} - p_{\textup{sv}} + p_{\textup{ref}} \right)\end{split}\]

_type = 'Synchronous_machine'

_name = 'Synchronous_machine_transient_model'

e_dprim

e_qprim

x_dprim

x_qprim

T_dprim

T_qprim

input_current(dae: hermess.system.Dae) → Tuple[casadi.SX, casadi.SX]

Parameters:

dae (hermess.system.Dae)

Return type:

Tuple[casadi.SX, casadi.SX]

electromagnetic(dae: hermess.system.Dae)

Two-axis (transient) electromagnetic model: stator currents, the e’_d/e’_q dynamics, and the air-gap power. The swing equation and orchestration live in the base (rotor / fgcall template).

Parameters:

dae (hermess.system.Dae)

class hermess.devices.synchronous.SynchronousSubtransient(avr=None, governor=None, pss=None, shaft=None)

Bases: Synchronous

Subtransient Anderson Fouad SG with TGOV1 governor and IEEEDC1A AVR The subtransient behavior of the synchronous generator is described by the following differential equations:

Rotor Dynamics

\[\begin{split}\dot{\delta} &= 2 \pi f_n \Delta \omega \\ \Delta \dot{\omega} &= \frac{1}{2 H} \Big( P_m - E_{d}^{\prime\prime} I_d - E_{q}^{\prime\prime} I_q + (X_q^{\prime\prime} - X_d^{\prime\prime}) I_d I_q - D \Delta \omega - f (\Delta \omega + 1) \Big) \\\end{split}\]

Electromagnetic Equations

\[\begin{split}\dot{E}_q^{\prime} &= \frac{1}{T_{d}^{\prime}} \Big( -E_q + E_f + (X_d - X_d^{\prime}) I_d \Big) \\ \dot{E}_d^{\prime} &= \frac{1}{T_{q}^{\prime}} \Big( -E_d - (X_q - X_q^{\prime}) I_q \Big) \\ \dot{E}_{q}^{\prime\prime} &= \frac{1}{T_{d}^{\prime\prime}} \Big( E_q - E_{q}^{\prime\prime} + (X_d^{\prime} - X_d^{\prime\prime}) I_d \Big) \\ \dot{E}_{d}^{\prime\prime} &= \frac{1}{T_{q}^{\prime\prime}} \Big( E_d - E_{d}^{\prime\prime} - (X_q^{\prime} - X_q^{\prime\prime}) I_q \Big) \\\end{split}\]

Excitation System Equations

\[\begin{split} \dot{E}_{\textup{fd}} &= \frac{1}{T_{E}} \left( -K_{E,i}E_{\textup{fd},i} + V_{R,i}\right)\\ \dot{R}_{f} &= \frac{1}{T_{F} } \left(- R_{f} + \frac{K_{F}}{T_{F}}E_{\textup{fd}} \right)\\ \dot{V}_{R} &= \frac{1}{T_{A}} \left(-V_{R} + K_{A}R_{f} - \frac{K_{A}K_{F}}{T_{F}}E_{\textup{fd}} + K_{A}(v_{\textup{ref}} - v_i) \right)\end{split}\]

Turbine-Governor System Equations

\[\begin{split}\dot{p}_{\textup{m}} &= \frac{1}{T_{ch}} \left( p_{\textup{sv}}- p_{\textup{m}} \right)\\ \dot{p}_{sv} &= \frac{1}{T_{sv}} \left( - \frac{\Delta \omega}{R_d} - p_{\textup{sv}} + p_{\textup{ref}} \right)\end{split}\]

_type = 'Synchronous_machine'

_name = 'Synchronous_machine_subtransient_model'

e_dprim

e_qprim

e_dsec

e_qsec

x_dprim

x_qprim

T_dprim

T_qprim

x_dsec

x_qsec

T_dsec

T_qsec

input_current(dae: hermess.system.Dae) → Tuple[casadi.SX, casadi.SX]

Parameters:

dae (hermess.system.Dae)

Return type:

Tuple[casadi.SX, casadi.SX]

electromagnetic(dae: hermess.system.Dae)

Anderson-Fouad (subtransient) electromagnetic model: stator currents, the e’/e’’ dynamics, and the air-gap power. Swing/orchestration in base.

Parameters:

dae (hermess.system.Dae)

class hermess.devices.synchronous.SynchronousSubtransientSP(avr=None, governor=None, pss=None, shaft=None)

Bases: Synchronous

Subtransient Sauer and Pai SG model with stator dynamics with TGOV1 governor and IEEEDC1A AVR

The model includes the following equations for rotor dynamics, stator dynamics, and the excitation system:

Rotor Dynamics

\[\begin{split}\dot{\delta} &= 2 \pi f_n \Delta \omega \\ \Delta \dot{\omega} &= \frac{1}{2 H} \Big( P_m - (\psi_d I_q - \psi_q I_d) - D \Delta \omega - f (\Delta \omega + 1) \Big)\end{split}\]

Electromagnetic Equations

The stator dynamics include the following equations for the flux linkages in the d and q axes:

\[\begin{split}\dot{E}_d' &= \frac{1}{T_q'} \Big( -E_d' + (X_q - X_q') (i_q - g_{q2} \Psi_{q2} - (1 - g_{q1}) i_q - g_{q2} E_d') \Big) \\ \dot{E}_q' &= \frac{1}{T_d'} \Big( -E_q' - (X_d - X_d') (i_d - g_{d2} \Psi_{d2} - (1 - g_{d1}) i_d + g_{d2} E_q') + E_f \Big)\\ \dot{\Psi}_{d2} &= \frac{1}{T_{d2}} \Big( -\Psi_{d2} + E_q' - (X_d' - X_l) i_d \Big) \\ \dot{\Psi}_{q2} &= \frac{1}{T_{q2}} \Big( -\Psi_{q2} - E_d' - (X_q' - X_l) i_q \Big)\end{split}\]

Flux Linkage Dynamics

The following equations describe the stator flux linkage dynamics in the d and q axes:

\[\begin{split}\dot{\Psi}_d &= 2 \pi f_n (R_s i_d + (1 + \Delta \omega) \Psi_q + v_d) \\ \dot{\Psi}_q &= 2 \pi f_n (R_s i_q - (1 + \Delta \omega) \Psi_d + v_q)\end{split}\]

Algebraic Equations

The following algebraic equations govern the system:

\[\begin{split}i_d &= \frac{1}{x_d''} \Big( -\psi_d + g_{d1} e_q' + (1 - g_{d1}) \psi_{d2} \Big)\\ i_q &= \frac{1}{x_q''} \Big( -\psi_q - g_{q1} e_d' + (1 - g_{q1}) \psi_{q2} \Big)\\ g_{d1} &= \frac{x_d'' - x_l}{x_d' - x_l}\\ g_{q1} &= \frac{x_q'' - x_l}{x_q' - x_l}\\ g_{d2} &= \frac{1 - g_{d1}}{x_d' - x_l}\\ g_{q2} &= \frac{1 - g_{q1}}{x_q' - x_l}\end{split}\]

_type = 'Synchronous_machine'

_name = 'Synchronous_machine_subtransient_model_Sauer_Pai'

e_dprim

e_qprim

psid

psiq

psid2

psiq2

x_l

gd1

gq1

gd2

gq2

x_dprim

x_qprim

T_dprim

T_qprim

x_dsec

x_qsec

T_dsec

T_qsec

sauer_pai(dae: hermess.system.Dae, i_d: casadi.SX, i_q: casadi.SX)

Sauer and Pai model. :param dae: :type dae: DAE :param i_d: :type i_d: casadi.SX :param i_q: :type i_q: casadi.SX

Parameters:

• dae (hermess.system.Dae)

• i_d (casadi.SX)

• i_q (casadi.SX)

_gd_coeffs() → None

Subtransient coupling coefficients used by both the stator-current expressions and the flux ODEs.

Return type:

None

input_current(dae: hermess.system.Dae) → Tuple[casadi.SX, casadi.SX]

Stator currents as inlined explicit expressions of the flux states.

Parameters:

dae (hermess.system.Dae)

Return type:

Tuple[casadi.SX, casadi.SX]

electromagnetic(dae: hermess.system.Dae)

Sauer-Pai (subtransient, stator dynamics) electromagnetic model. The stator-current derivation is input_current() (overridden by the _DAE variant to expose i_d/i_q as private algebraics); the shared flux/ stator physics and the air-gap power live in sauer_pai().

Parameters:

dae (hermess.system.Dae)

class hermess.devices.synchronous.SynchronousSubtransientSP6(avr=None, governor=None, pss=None, shaft=None)

Bases: Synchronous

Subtransient Sauer and Pai SG 6th order model with neglected stator dynamics (stator modeled with algebraic equations) and included TGOV1 governor and IEEEDC1A AVR

The model includes the following equations:

Rotor Dynamics

\[\begin{split}\dot{\delta} &= 2 \pi f_n \Delta \omega \\ \Delta \dot{\omega} &= \frac{1}{2 H} \Big( P_m - (\psi_d I_q - \psi_q I_d) - D \Delta \omega - f (\Delta \omega + 1) \Big)\end{split}\]

Electromagnetic Equations

The stator dynamics include the following equations for the flux linkages in the d and q axes:

\[\begin{split}\dot{E}_d' &= \frac{1}{T_q'} \Big( -E_d' + (X_q - X_q') (i_q - g_{q2} \Psi_{q2} - (1 - g_{q1}) i_q - g_{q2} E_d') \Big) \\ \dot{E}_q' &= \frac{1}{T_d'} \Big( -E_q' - (X_d - X_d') (i_d - g_{d2} \Psi_{d2} - (1 - g_{d1}) i_d + g_{d2} E_q') + E_f \Big)\\ \dot{\Psi}_{d2} &= \frac{1}{T_{d2}} \Big( -\Psi_{d2} + E_q' - (X_d' - X_l) i_d \Big) \\ \dot{\Psi}_{q2} &= \frac{1}{T_{q2}} \Big( -\Psi_{q2} - E_d' - (X_q' - X_l) i_q \Big)\end{split}\]

Flux Linkage Dynamics

The following equations describe the stator flux linkage dynamics in the d and q axes:

Algebraic Equations

The following algebraic equations govern the system:

\[\begin{split}0 &=-i_d +\frac{1}{x_d''} \Big( -\psi_d + g_{d1} e_q' + (1 - g_{d1}) \psi_{d2} \Big)\\ 0&=-i_q +\frac{1}{x_q''} \Big( -\psi_q - g_{q1} e_d' + (1 - g_{q1}) \psi_{q2} \Big)\\ 0&= R_s i_d + (1 + \Delta \omega) \Psi_q + v_d \\ 0&= R_s i_q - (1 + \Delta \omega) \Psi_d + v_q \\ g_{d1} &= \frac{x_d'' - x_l}{x_d' - x_l}\\ g_{q1} &= \frac{x_q'' - x_l}{x_q' - x_l}\\ g_{d2} &= \frac{1 - g_{d1}}{x_d' - x_l}\\ g_{q2} &= \frac{1 - g_{q1}}{x_q' - x_l}\end{split}\]

_type = 'Synchronous_machine'

_name = 'Synchronous_machine_subtransient_model_Sauer_Pai_6th_order'

e_dprim

e_qprim

psid2

psiq2

x_l

gd1

gq1

gd2

gq2

x_dprim

x_qprim

T_dprim

T_qprim

x_dsec

x_qsec

T_dsec

T_qsec

sauer_pai_6(dae: hermess.system.Dae, i_d: casadi.SX, i_q: casadi.SX, psid: casadi.SX, psiq: casadi.SX)

Sauer and Pai model. :param dae: :type dae: DAE :param i_d: :type i_d: casadi.SX :param i_q: :type i_q: casadi.SX :param psid: :type psid: casadi.SX :param psiq: :type psiq: casadi.SX

Returns:

• Args – psid (): psiq ():

Parameters:

• dae (hermess.system.Dae)

• i_d (casadi.SX)

• i_q (casadi.SX)

• psid (casadi.SX)

• psiq (casadi.SX)

stator_solve(dae: hermess.system.Dae)

Eliminate the four algebraic stator unknowns (i_d, i_q, psi_d, psi_q) by a symbolic 4x4 solve per machine. Returns (i_d, i_q, psid, psiq). The _DAE variant overrides this to expose them as private algebraics.

Parameters:

dae (hermess.system.Dae)

electromagnetic(dae: hermess.system.Dae)

Sauer-Pai 6th-order (algebraic stator) electromagnetic model. The four stator unknowns come from stator_solve() (overridden by the _DAE variant to expose them as private algebraics); the shared flux physics and the air-gap power live in sauer_pai_6().

Parameters:

dae (hermess.system.Dae)

class hermess.devices.synchronous.SynchronousSubtransientSP6DAE(avr=None, governor=None, pss=None, shaft=None)

Bases: SynchronousSubtransientSP6

Subtransient Sauer-Pai 6th-order machine expressed as an explicit DAE.

Identical physics to SynchronousSubtransientSP6, but the four stator unknowns (i_d, i_q, psi_d, psi_q) are declared as device-private algebraic variables/equations (_algebs_int) and handed to the DAE solver rather than symbolically eliminated with a per-instance ca.solve. Serves as a parity vehicle for the private-algebraic mechanism: it reproduces SynchronousSubtransientSP6 to integrator tolerance.

The stator block is linear in the unknowns, so the parent’s ca.solve is the exact solution of g = 0; the DAE integrator converges to the same values via a numerical Newton solve per step instead of a closed-form expression built once.

_name = 'Synchronous_machine_subtransient_model_Sauer_Pai_6th_order_DAE'

stator_solve(dae: hermess.system.Dae)

The four stator unknowns are device-private algebraic variables in dae.y; write their defining equations and return them, so the integrator solves the 4x4 block instead of ca.solve. Inherits SP6’s electromagnetic() and the base orchestration.

Parameters:

dae (hermess.system.Dae)

class hermess.devices.synchronous.SynchronousSubtransientSP_DAE(avr=None, governor=None, pss=None, shaft=None)

Bases: SynchronousSubtransientSP

Subtransient Sauer-Pai machine (with stator dynamics) with the stator currents exposed as explicit device-private algebraic variables.

Identical physics to SynchronousSubtransientSP, but the stator currents i_d, i_q are declared as private algebraic variables (_algebs_int) with their defining equations 0 = -i + <expr> handed to the DAE solver, rather than inlined as explicit expressions of the flux states. Reproduces SynchronousSubtransientSP to integrator tolerance.

A parity vehicle for the private-algebraic mechanism: the parent model initialises robustly and the defining equations are linear with a trivially non-singular Jacobian (index-1).

_name = 'Synchronous_machine_subtransient_model_Sauer_Pai_DAE'

input_current(dae: hermess.system.Dae) → Tuple[casadi.SX, casadi.SX]

Stator currents as device-private algebraic variables in dae.y, defined by 0 = -i + <explicit expression> (the same expression the parent inlines). The defining Jacobian dg/di = -1 is trivially non-singular (index-1). Inherits SP’s electromagnetic() and the base orchestration.

Parameters:

dae (hermess.system.Dae)

Return type:

Tuple[casadi.SX, casadi.SX]

class hermess.devices.synchronous.GENROU(avr=None, governor=None, pss=None, shaft=None)

Bases: Synchronous

Sixth-order round-rotor synchronous machine (PSS/E GENROU): two rotor windings per axis, in the K-coefficient form of Gibbard & Vowles 2014, Appendix I.5.1, eqs. (11)-(24). A general-purpose machine model (the SEA benchmark is one user). States: \(\delta, \omega, E'_q, \psi_{kd}, E'_d, \psi_{kq}\); the stator is algebraic with the transformer voltages and the speed dependency of the rotational voltages neglected, and magnetic saturation disabled:

\[\begin{split}T'_{d0}\dot{E}'_q &= E_{fd} - [K_{1d}E'_q + K_{2d}\psi_{kd} + K_{3d}I_d] \\ T''_{d0}\dot{\psi}_{kd} &= E'_q - \psi_{kd} - (X'_d - X_a) I_d \\ T'_{q0}\dot{E}'_d &= K_{1q}E'_d + K_{2q}\psi_{kq} + K_{3q}I_q \\ T''_{q0}\dot{\psi}_{kq} &= E'_d - \psi_{kq} + (X'_q - X_a) I_q\end{split}\]

with the stator equations \(v_d = -R_s I_d + X''_q I_q + \psi''_q\), \(v_q = -R_s I_q - X''_d I_d + \psi''_d\), the subtransient flux linkages (19)-(20) and the coupling coefficients (22)/(24). The air-gap power follows from the stator equations as \(P_e = \psi''_d I_q + \psi''_q I_d + (X''_q - X''_d) I_d I_q\). The +(X'_q - X_a) I_q sign in the last differential equation is deliberate (see the sign note in electromagnetic()): with the stator convention above, the q-axis steady state must reproduce v_d = X_q I_q. It is independently cross-validated against the PSID GENROU reference equations. All reactances/time constants in per unit on the machine MVA rating.

_type = 'Synchronous_machine'

_name = 'Synchronous_machine_GENROU'

e_qprim

psi_kd

e_dprim

psi_kq

x_a

x_dprim

x_qprim

x_dsec

x_qsec

T_d0prim

T_q0prim

T_d0sec

T_q0sec

finit_guess(dae: hermess.system.Dae)

Operating-point-aware Newton seeds from the power-flow phasors.

The machine equations have a mirror (saddle) equilibrium at delta − π with all EMFs negated; with a uniform delta guess, Newton lands in the wrong basin for machines at buses with large voltage angles. Seed each instance from the standard phasor construction E_q = V + (R_s + j·x_q)·I instead (and the exact steady-state chain for the rotor states), which puts the guess next to the physical root.

Parameters:

dae (hermess.system.Dae)

_seed_values(dae, i, *, delta, eq1, psikd, ed1, psikq, iq, efd, v_t) → dict

Map the phasor-derived quantities onto this class’s state names; AVR/PSS/governor states default to their steady-state values.

Return type:

dict

_subtransient_flux(dae: hermess.system.Dae)

psi’’_d, psi’’_q from eqs. (19)-(20).

Parameters:

dae (hermess.system.Dae)

input_current(dae: hermess.system.Dae) → Tuple[casadi.SX, casadi.SX]

Parameters:

dae (hermess.system.Dae)

Return type:

Tuple[casadi.SX, casadi.SX]

electromagnetic(dae: hermess.system.Dae)

Subclass hook: define the stator currents (i_d, i_q), write the flux/emf differential equations, and return (i_d, i_q, Pe) where Pe is the air-gap power/torque consumed by rotor().

Parameters:

dae (hermess.system.Dae)

class hermess.devices.synchronous.GENSAL(avr=None, governor=None, pss=None, shaft=None)

Bases: GENROU

Fifth-order salient-pole synchronous machine (PSS/E GENSAL): two rotor windings on the d-axis, one on the q-axis. The d-axis circuits and the equations of motion are those of GENROU; the q-axis transient pair \((E'_d, \psi_{kq})\) is replaced by the single subtransient flux state \(\psi''_q\), in the K-coefficient form of Gibbard & Vowles 2014, Appendix I.5.2, eqs. (25)-(26):

\[T''_{q0}\,\dot{\psi}''_q = -\psi''_q + (X_q - X''_q) I_q .\]

The +(X_q - X''_q) I_q sign is deliberate (the q-axis steady state must give \(\psi''_q = (X_q - X''_q) I_q\) so that v_d = X_q I_q); it is cross-validated against the PSID GENSAL reference equations, which define their psi_q'' state with the opposite sign.

_name = 'Synchronous_machine_GENSAL'

psi_qsec

_subtransient_flux(dae: hermess.system.Dae)

psi’’_d, psi’’_q from eqs. (19)-(20).

Parameters:

dae (hermess.system.Dae)

electromagnetic(dae: hermess.system.Dae)

Subclass hook: define the stator currents (i_d, i_q), write the flux/emf differential equations, and return (i_d, i_q, Pe) where Pe is the air-gap power/torque consumed by rotor().

Parameters:

dae (hermess.system.Dae)