Automatic voltage regulators#

The AVR strategy is plugged onto a machine with the avr keyword, for example avr = "SEXST". It reads the terminal voltage and the stabilizing signal \(V_s\) of the PSS strategy and produces the field voltage \(E_{fd}\).

`avr =`	Class	Description
`"AVRAC1A"`	`AVRAC1A`	IEEE Std 421.5 AC1A exciter (small-signal form, no limits or saturation), as used by the 14-generator South East Australian benchmark (Gibbard and Vowles 2014, Fig. 21, Tables 16 and 27).
`"AVRCONST"`	`AVRCONST`	Constant field voltage: no excitation dynamics, the zero-response limit of any exciter (\(K_A \to 0\) with the operating point held).
`"AVRKundur"`	`AVRKundur`	Kundur two-area AVR: transducer plus lead-lag (transient gain reduction), with the field voltage as an algebraic output (Kundur, Power System Stability and Control, 1994, two-area example).
`"AVRKundur_Filter"`	`AVRKundur_Filter`	`AVRKundur` with an additional first-order filter on the exciter output, which removes the algebraic feedthrough (all states differential) and damps unrealistically fast dynamics.
`"AVRKundur_NoTGR"`	`AVRKundur_NoTGR`	`AVRKundur` with the transient gain reduction removed: a plain high-gain static exciter with only a terminal-voltage transducer.
`"AVRKundur_ODE"`	`AVRKundur_ODE`	All-ODE realization of the Kundur two-area AVR (transducer plus lead-lag with transient gain reduction), with the lead realized through the derivative of the measurement so that \(E_{fd}\) stays a differential state.
`"AVRST1A"`	`AVRST1A`	IEEE Std 421.5 ST1A static exciter (small-signal form, no limits), as used by the 14-generator South East Australian benchmark (Gibbard and Vowles 2014, Fig. 20, Tables 16 and 26).
`"AVRSimple"`	`AVRSimple`	Simple proportional exciter: the field voltage is a pure integrator on the voltage error (the PSID `AVRSimple`).
`"IEEEDC1A"`	`IEEEDC1A`	IEEE DC1A rotating exciter with stabilizing rate feedback (P.
`"SEXST"`	`SEXST`	Simplified static exciter: a single first-order lag on the voltage error.

SEXST block diagram — SEXST: first-order static exciter.#

class hermess.devices.avr.SEXST[source]

Simplified static exciter: a single first-order lag on the voltage error.

Selected on a machine line with avr = "SEXST".

Model.

\[T_E \, \dot{E}_{fd} = -E_{fd} + K_A \left( V_{ref} - V_t + V_s \right)\]

with \(V_t = |\bar{v}_n|\) the terminal-voltage magnitude and \(V_s\) the stabilizer signal (0 when no PSS is attached).

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	voltage regulator gain	200
`TE`	\(T_E\)	exciter time constant [s]	0.1
`Efd_max` / `Efd_min`	\(E_{fd}^{max,min}\)	field-voltage limits (with `incl_lim`)	5 / 0
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

IEEE DC1A block diagram — IEEE DC1A: regulator, rotating exciter and stabilizing rate feedback.#

class hermess.devices.avr.IEEEDC1A[source]

IEEE DC1A rotating exciter with stabilizing rate feedback (P. W. Sauer and M. A. Pai, Power System Dynamics and Stability, p. 100).

Selected on a machine line with avr = "IEEEDC1A".

Model.

\[\begin{split}T_E \, \dot{E}_{fd} &= -K_E \, E_{fd} + V_r \\ T_F \, \dot{R}_f &= -R_f + \frac{K_F}{T_F} E_{fd} \\ T_A \, \dot{V}_r &= -V_r + K_A \Bigl( R_f - \frac{K_F}{T_F} E_{fd} + V_{ref} - V_t + V_s \Bigr)\end{split}\]

with \(V_t = |\bar{v}_n|\) and \(V_s\) the stabilizer signal. With incl_lim the \(V_r\) equation is multiplied by the anti-windup switch that freezes the regulator at \(V_r^{max,min}\).

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	voltage regulator gain	200
`TA`	\(T_A\)	voltage regulator time constant [s]	0.015
`KF`	\(K_F\)	rate-feedback (stabilizer) gain	1
`TF`	\(T_F\)	rate-feedback time constant [s]	0.1
`KE`	\(K_E\)	exciter field constant (no saturation)	1
`TE`	\(T_E\)	exciter time constant [s]	0.04
`Vr_max` / `Vr_min`	\(V_r^{max,min}\)	regulator limits (with `incl_lim`)	5 / 0
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Rf`	\(R_f\)	rate-feedback state [p.u.]
`Vr`	\(V_r\)	pilot-exciter (regulator) voltage (state) [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

ST1A block diagram — IEEE ST1A: transducer, two lead-lags and the regulator lag.#

class hermess.devices.avr.AVRST1A[source]

IEEE Std 421.5 ST1A static exciter (small-signal form, no limits), as used by the 14-generator South East Australian benchmark (Gibbard and Vowles 2014, Fig. 20, Tables 16 and 26).

Selected on a machine line with avr = "AVRST1A".

Model. Transducer, two lead-lags (each realized as a lag state plus direct feedthrough), and the regulator lag:

\[\begin{split}T_r \, \dot{V}_{tr} &= V_t - V_{tr}, \qquad e = V_{ref} - V_{tr} + V_s \\ T_B \, \dot{V}_{ll1} &= e - V_{ll1}, \qquad y_1 = \Bigl(1 - \frac{T_C}{T_B}\Bigr) V_{ll1} + \frac{T_C}{T_B}\, e \\ T_{B1} \, \dot{V}_{ll2} &= y_1 - V_{ll2}, \qquad y_2 = \Bigl(1 - \frac{T_{C1}}{T_{B1}}\Bigr) V_{ll2} + \frac{T_{C1}}{T_{B1}}\, y_1 \\ T_A \, \dot{E}_{fd} &= K_A \, y_2 - E_{fd}\end{split}\]

with \(V_t = |\bar{v}_n|\) and \(V_s\) the stabilizer signal. \(T_B\) and \(T_{B1}\) must be positive; a unity block is obtained exactly with \(T_C = T_B\), and a transducer given as \(T_r = 0\) in the data is approximated by a small lag (\(10^{-4}\) s, a parasitic pole far above the rotor-mode range).

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	regulator gain	300
`TA`	\(T_A\)	regulator time constant [s]	0.05
`Tr`	\(T_r\)	voltage transducer time constant [s]	1e-4
`TB`	\(T_B\)	first lead-lag denominator time constant [s]	1
`TC`	\(T_C\)	first lead-lag numerator time constant [s]	1
`TB1`	\(T_{B1}\)	second lead-lag denominator time constant [s]	1e-4
`TC1`	\(T_{C1}\)	second lead-lag numerator time constant [s]	1e-4
`Vtr`	\(V_{tr}\)	transducer state [p.u.]
`Vll1`	\(V_{ll1}\)	first lead-lag lag state [p.u.]
`Vll2`	\(V_{ll2}\)	second lead-lag lag state [p.u.]
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

AC1A block diagram — IEEE AC1A: regulator, rotating exciter and washout rate feedback.#

class hermess.devices.avr.AVRAC1A[source]

IEEE Std 421.5 AC1A exciter (small-signal form, no limits or saturation), as used by the 14-generator South East Australian benchmark (Gibbard and Vowles 2014, Fig. 21, Tables 16 and 27). Exciter saturation, armature reaction and rectifier regulation are neglected (\(K_C = K_D = 0\)), and the lead-lag is unity.

Selected on a machine line with avr = "AVRAC1A".

Model. Washout realization of the rate feedback \(s K_F/(1 + s T_F)\), regulator lag, and rotating exciter:

\[\begin{split}V_f &= \frac{K_F}{T_F} E_{fd} - V_{fb}, \qquad T_F \, \dot{V}_{fb} = V_f \\ T_A \, \dot{V}_r &= K_A \left( V_{ref} - V_t + V_s - V_f \right) - V_r \\ T_E \, \dot{E}_{fd} &= V_r - K_E \, E_{fd}\end{split}\]

with \(V_t = |\bar{v}_n|\) (unfiltered; the benchmark data has \(T_r = 0\)) and \(V_s\) the stabilizer signal.

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	regulator gain	400
`TA`	\(T_A\)	regulator time constant [s]	0.02
`KE`	\(K_E\)	exciter constant	1
`TE`	\(T_E\)	exciter time constant [s]	1
`KF`	\(K_F\)	rate feedback gain	0.03
`TF`	\(T_F\)	rate feedback time constant [s]	1
`Vr`	\(V_r\)	regulator output (state) [p.u.]
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Vfb`	\(V_{fb}\)	rate-feedback filter state [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

Kundur AVR family block diagram — The Kundur two-area AVR family: transducer, gain and transient-gain-reduction lead-lag, in four realizations.#

class hermess.devices.avr.AVRKundur[source]

Kundur two-area AVR: transducer plus lead-lag (transient gain reduction), with the field voltage as an algebraic output (Kundur, Power System Stability and Control, 1994, two-area example).

Selected on a machine line with avr = "AVRKundur".

Model. The lead-lag \(E_{fd} = K_A \frac{1 + s T_A}{1 + s T_B} (V_{ref} - V_{tr} + V_s)\) is proper but not strictly proper, so its output has a direct feedthrough and is genuinely algebraic. It is realized as one lag-pole state \(V_l\) plus the algebraic output:

\[\begin{split}T_R \, \dot{V}_{tr} &= -V_{tr} + V_t \\ T_B \, \dot{V}_l &= -V_l + K_A \left( V_{ref} - V_{tr} + V_s \right) \\ 0 &= -E_{fd} + \Bigl(1 - \frac{T_A}{T_B}\Bigr) V_l + \frac{T_A}{T_B} K_A \left( V_{ref} - V_{tr} + V_s \right)\end{split}\]

with \(V_t = |\bar{v}_n|\) and \(V_s\) the stabilizer signal. The third line is the lead feedthrough \(D = T_A/T_B\). Efd is exposed via algebs() (not states()) and rides the device-private-algebraic mechanism; the host machine reads it through Synchronous.var_sym('Efd').

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	voltage regulator gain	200
`TA`	\(T_A\)	lead time constant [s]	1
`TB`	\(T_B\)	lag time constant [s]	10
`TR`	\(T_R\)	transducer time constant [s]	0.01
`Efd_max` / `Efd_min`	\(E_{fd}^{max,min}\)	field-voltage limits (with `incl_lim`)	5 / 0
`Vl`	\(V_l\)	lag-pole state [p.u.]
`Vtr`	\(V_{tr}\)	transducer state [p.u.]
`Efd`	\(E_{fd}\)	field voltage (private algebraic) [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

class hermess.devices.avr.AVRKundur_ODE[source]

All-ODE realization of the Kundur two-area AVR (transducer plus lead-lag with transient gain reduction), with the lead realized through the derivative of the measurement so that \(E_{fd}\) stays a differential state.

Selected on a machine line with avr = "AVRKundur_ODE".

Model.

\[\begin{split}T_R \, \dot{V}_{tr} &= -V_{tr} + V_t \\ T_B \, \dot{E}_{fd} &= -E_{fd} + K_A \left( V_{ref} - V_{tr} + V_s \right) - K_A T_A \, \dot{V}_{tr}\end{split}\]

with \(V_t = |\bar{v}_n|\), \(V_s\) the stabilizer signal, and \(\dot{V}_{tr} = (V_t - V_{tr})/T_R\) substituted symbolically. The setpoint-derivative term \(K_A T_A \dot{V}_{ref}\) is omitted; it is zero for a constant setpoint.

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	voltage regulator gain	200
`TA`	\(T_A\)	lead time constant [s]	0.015
`TB`	\(T_B\)	lag time constant [s]	0.02
`TR`	\(T_R\)	transducer time constant [s]	0.01
`Efd_max` / `Efd_min`	\(E_{fd}^{max,min}\)	field-voltage limits (with `incl_lim`)	5 / 0
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Vtr`	\(V_{tr}\)	transducer state [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

class hermess.devices.avr.AVRKundur_Filter[source]

AVRKundur with an additional first-order filter on the exciter output, which removes the algebraic feedthrough (all states differential) and damps unrealistically fast dynamics.

Selected on a machine line with avr = "AVRKundur_Filter".

Model.

\[\begin{split}T_R \, \dot{V}_{tr} &= -V_{tr} + V_t \\ T_B \, \dot{V}_l &= -V_l + K_A \, e \\ T_{fd} \, \dot{E}_{fd} &= -E_{fd} + V_l + \frac{T_A}{T_B} \left( K_A \, e - V_l \right)\end{split}\]

with the error \(e = V_{ref} - V_{tr} + V_s\), \(V_t = |\bar{v}_n|\), and \(V_s\) the stabilizer signal. The first two lines are the transducer and the lag pole of the lead-lag; the third passes the lead-lag output (lag state plus feedthrough \(T_A/T_B\)) through the output filter \(T_{fd}\).

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	voltage regulator gain	200
`TA`	\(T_A\)	lead time constant [s]	1
`TB`	\(T_B\)	lag time constant [s]	10
`TR`	\(T_R\)	transducer time constant [s]	0.01
`Tfd`	\(T_{fd}\)	exciter output filter time constant [s]	0.01
`Efd_max` / `Efd_min`	\(E_{fd}^{max,min}\)	field-voltage limits (with `incl_lim`)	5 / 0
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Vl`	\(V_l\)	lag-pole state [p.u.]
`Vtr`	\(V_{tr}\)	transducer state [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

class hermess.devices.avr.AVRKundur_NoTGR[source]

AVRKundur with the transient gain reduction removed: a plain high-gain static exciter with only a terminal-voltage transducer.

Selected on a machine line with avr = "AVRKundur_NoTGR".

Model.

\[\begin{split}T_R \, \dot{V}_{tr} &= -V_{tr} + V_t \\ 0 &= -E_{fd} + K_A \left( V_{ref} - V_{tr} + V_s \right)\end{split}\]

with \(V_t = |\bar{v}_n|\) and \(V_s\) the stabilizer signal. \(E_{fd}\) is an instantaneous gain on the transduced error, hence a private algebraic (read by the machine via Synchronous.var_sym('Efd')). With a high \(K_A\) and no transient gain reduction this exciter reduces the damping of the electromechanical modes, the classic setting in which a power system stabilizer is needed to restore damping.

Symbols.

Code	Symbol	Meaning	Default
`KA`	\(K_A\)	voltage regulator gain	200
`TR`	\(T_R\)	transducer time constant [s]	0.02
`Efd_max` / `Efd_min`	\(E_{fd}^{max,min}\)	field-voltage limits (with `incl_lim`)	5 / 0
`Vtr`	\(V_{tr}\)	transducer state [p.u.]
`Efd`	\(E_{fd}\)	field voltage (private algebraic) [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)

Two minimal exciters close the family, useful as limiting cases and for matching references without excitation dynamics. AVRCONST holds the field voltage constant at its operating point, and AVRSimple integrates the voltage error, \(\dot{E}_{fd} = K_v \left( V_{ref} - V_t \right)\).

class hermess.devices.avr.AVRCONST[source]

Constant field voltage: no excitation dynamics, the zero-response limit of any exciter (\(K_A \to 0\) with the operating point held). The excitation-system counterpart of the constant-power governor GOVCONST.

Selected on a machine line with avr = "AVRCONST".

Model.

\[0 = -E_{fd} + V_{ref}\]

\(V_{ref}\) is solved by the initialization to the field voltage that holds the power-flow operating point, and stays frozen afterwards. Used by validation cases that deliberately exclude excitation dynamics, e.g. the cross-tool machine benchmarks against ANDES, whose machines hold \(v_f\) constant when no exciter is attached.

Symbols.

Code	Symbol	Meaning	Default
`Efd`	\(E_{fd}\)	field voltage (private algebraic) [p.u.]
`Vf_ref`	\(V_{ref}\)	field-voltage setpoint (set by the initialization)

class hermess.devices.avr.AVRSimple[source]

Simple proportional exciter: the field voltage is a pure integrator on the voltage error (the PSID AVRSimple).

Selected on a machine line with avr = "AVRSimple".

Model.

\[\dot{E}_{fd} = K_v \left( V_{ref} - V_t \right)\]

with \(V_t = |\bar{v}_n|\) the terminal-voltage magnitude. The integrator makes the steady-state voltage error exactly zero, so the initialization pins \(V_{ref} = V_t\).

Symbols.

Code	Symbol	Meaning	Default
`Kv`	\(K_v\)	integrator gain	1
`Efd`	\(E_{fd}\)	field voltage (state) [p.u.]
`Vf_ref`	\(V_{ref}\)	voltage setpoint (set by the initialization)