pydynamicestimator.devices.synchronous

Classes

`Synchronous`	Metaclass for SG in rectangular coordinates with TGOV1 governor and IEEEDC1 exciter
`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

Module Contents

class pydynamicestimator.devices.synchronous.Synchronous[source]

Bases: pydynamicestimator.devices.device.DeviceRect

Metaclass for SG in rectangular coordinates with TGOV1 governor and IEEEDC1 exciter

fn

H

R_s

x_d

x_q

D

f

Rd

Tch

Tsv

KA

TA

KF

TF

KE

TE

Vr_max

Vr_min

psv_max

psv_min

ns = 7

delta

omega

psv

pm

Efd

Rf

Vr

Vf_ref

Pref

gcall(dae: pydynamicestimator.system.Dae, i_d: casadi.SX, i_q: casadi.SX) → None[source]

Parameters:

dae (pydynamicestimator.system.Dae)
i_d (casadi.SX)
i_q (casadi.SX)

Return type:

None

tgov1(dae: pydynamicestimator.system.Dae) → None[source]

TGOV1 governor model as presented in Power System Dynamics and Stability by P.W. Sauer and M.A. Pai, 2006. (page 100)

Parameters:: dae (DAE)
Return type:: None

ieeedc1a(dae: pydynamicestimator.system.Dae) → None[source]

IEEEDC1 exciter and AVR model as presented in Power System Dynamics and Stability by P.W. Sauer and M.A. Pai, 2006. (page 100)

Parameters:: dae (differential-algebraic model class)
Return type:: None

class pydynamicestimator.devices.synchronous.SynchronousTransient[source]

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: pydynamicestimator.system.Dae) → Tuple[casadi.SX, casadi.SX][source]

Parameters:: dae (pydynamicestimator.system.Dae)
Return type:: Tuple[casadi.SX, casadi.SX]

two_axis(dae, i_d: casadi.SX, i_q: casadi.SX) → None[source]

Mechanical and electrical equations for the two-axis model.

Parameters:

dae – Dae
i_d (casadi.SX) – Stator d-current
i_q (casadi.SX) – Stator q-current

Returns:

None

Return type:

None

fgcall(dae: pydynamicestimator.system.Dae) → None[source]

A method that executes the differential and algebraic equations of the model and adds them to the appropriate places in the Dae class

Parameters:: dae – an instance of a class Dae
Returns:: None
Return type:: None

class pydynamicestimator.devices.synchronous.SynchronousSubtransient[source]

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: pydynamicestimator.system.Dae) → Tuple[casadi.SX, casadi.SX][source]

Parameters:: dae (pydynamicestimator.system.Dae)
Return type:: Tuple[casadi.SX, casadi.SX]

anderson_fouad(dae: pydynamicestimator.system.Dae, i_d: casadi.SX, i_q: casadi.SX)[source]

Parameters:

dae
i_d
i_q

Returns:

Return type:

fgcall(dae: pydynamicestimator.system.Dae) → None[source]

A method that executes the differential and algebraic equations of the model and adds them to the appropriate places in the Dae class

Parameters:: dae – an instance of a class Dae
Returns:: None
Return type:: None

class pydynamicestimator.devices.synchronous.SynchronousSubtransientSP[source]

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: pydynamicestimator.system.Dae, i_d: casadi.SX, i_q: casadi.SX)[source]

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 (pydynamicestimator.system.Dae)
i_d (casadi.SX)
i_q (casadi.SX)

fgcall(dae: pydynamicestimator.system.Dae) → None[source]

A method that executes the differential and algebraic equations of the model and adds them to the appropriate places in the Dae class

Parameters:: dae – an instance of a class Dae
Returns:: None
Return type:: None

class pydynamicestimator.devices.synchronous.SynchronousSubtransientSP6[source]

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: pydynamicestimator.system.Dae, i_d: casadi.SX, i_q: casadi.SX, psid: casadi.SX, psiq: casadi.SX)[source]

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 (pydynamicestimator.system.Dae)
i_d (casadi.SX)
i_q (casadi.SX)
psid (casadi.SX)
psiq (casadi.SX)

fgcall(dae: pydynamicestimator.system.Dae) → None[source]

A method that executes the differential and algebraic equations of the model and adds them to the appropriate places in the Dae class

Parameters:: dae – an instance of a class Dae
Returns:: None
Return type:: None