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

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

Parameters:

• dae (pydynamicestimator.system.Dae)

• i_d (casadi.SX)

• i_q (casadi.SX)

Return type:

None

tgov1(dae: pydynamicestimator.system.Dae) → None

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

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

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]

Parameters:

dae (pydynamicestimator.system.Dae)

Return type:

Tuple[casadi.SX, casadi.SX]

two_axis(dae, i_d: casadi.SX, i_q: casadi.SX) → None

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

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

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]

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)

Parameters:

• dae

• i_d

• i_q

Returns:

Return type:

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

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

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)

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

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

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)

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

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