pydynamicestimator.devices.synchronous
======================================

.. py:module:: pydynamicestimator.devices.synchronous


Classes
-------

.. autoapisummary::

   pydynamicestimator.devices.synchronous.Synchronous
   pydynamicestimator.devices.synchronous.SynchronousTransient
   pydynamicestimator.devices.synchronous.SynchronousSubtransient
   pydynamicestimator.devices.synchronous.SynchronousSubtransientSP
   pydynamicestimator.devices.synchronous.SynchronousSubtransientSP6


Module Contents
---------------

.. py:class:: Synchronous

   Bases: :py:obj:`pydynamicestimator.devices.device.DeviceRect`


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


   .. py:attribute:: fn


   .. py:attribute:: H


   .. py:attribute:: R_s


   .. py:attribute:: x_d


   .. py:attribute:: x_q


   .. py:attribute:: D


   .. py:attribute:: f


   .. py:attribute:: Rd


   .. py:attribute:: Tch


   .. py:attribute:: Tsv


   .. py:attribute:: KA


   .. py:attribute:: TA


   .. py:attribute:: KF


   .. py:attribute:: TF


   .. py:attribute:: KE


   .. py:attribute:: TE


   .. py:attribute:: Vr_max


   .. py:attribute:: Vr_min


   .. py:attribute:: psv_max


   .. py:attribute:: psv_min


   .. py:attribute:: ns
      :value: 7



   .. py:attribute:: delta


   .. py:attribute:: omega


   .. py:attribute:: psv


   .. py:attribute:: pm


   .. py:attribute:: Efd


   .. py:attribute:: Rf


   .. py:attribute:: Vr


   .. py:attribute:: Vf_ref


   .. py:attribute:: Pref


   .. py:method:: gcall(dae: pydynamicestimator.system.Dae, i_d: casadi.SX, i_q: casadi.SX) -> None


   .. py:method:: 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)

      :param dae:
      :type dae: DAE



   .. py:method:: 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)

      :param dae:
      :type dae: differential-algebraic model class



.. py:class:: SynchronousTransient

   Bases: :py:obj:`Synchronous`


   Transient two-axis SG with TGOV1 governor and IEEEDC1A AVR

   **Rotor Dynamics**


   .. math::

       \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)

   **Electromagnetic Equations**


   .. math::

       \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)

   **Excitation System Equations**


   .. math::

        \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)

   **Turbine-Governor System Equations**


   .. math::

       \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)






   .. py:attribute:: _type
      :value: 'Synchronous_machine'



   .. py:attribute:: _name
      :value: 'Synchronous_machine_transient_model'



   .. py:attribute:: e_dprim


   .. py:attribute:: e_qprim


   .. py:attribute:: x_dprim


   .. py:attribute:: x_qprim


   .. py:attribute:: T_dprim


   .. py:attribute:: T_qprim


   .. py:method:: input_current(dae: pydynamicestimator.system.Dae) -> Tuple[casadi.SX, casadi.SX]


   .. py:method:: two_axis(dae, i_d: casadi.SX, i_q: casadi.SX) -> None

      Mechanical and electrical equations for the two-axis model.

      :param dae: Dae

      :type dae:

      :param i_d: Stator d-current

      :type i_d: casadi.SX

      :param i_q: Stator q-current

      :type i_q: casadi.SX

      :return: None

      :rtype: None



   .. py:method:: 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

      :param dae: an instance of a class Dae

      :type dae:

      :return: None

      :rtype: None



.. py:class:: SynchronousSubtransient

   Bases: :py:obj:`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**


   .. math::

       \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) \\

   **Electromagnetic Equations**


   .. math::

       \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) \\

   **Excitation System Equations**


   .. math::

        \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)

   **Turbine-Governor System Equations**


   .. math::

       \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)


   .. py:attribute:: _type
      :value: 'Synchronous_machine'



   .. py:attribute:: _name
      :value: 'Synchronous_machine_subtransient_model'



   .. py:attribute:: e_dprim


   .. py:attribute:: e_qprim


   .. py:attribute:: e_dsec


   .. py:attribute:: e_qsec


   .. py:attribute:: x_dprim


   .. py:attribute:: x_qprim


   .. py:attribute:: T_dprim


   .. py:attribute:: T_qprim


   .. py:attribute:: x_dsec


   .. py:attribute:: x_qsec


   .. py:attribute:: T_dsec


   .. py:attribute:: T_qsec


   .. py:method:: input_current(dae: pydynamicestimator.system.Dae) -> Tuple[casadi.SX, casadi.SX]


   .. py:method:: anderson_fouad(dae: pydynamicestimator.system.Dae, i_d: casadi.SX, i_q: casadi.SX)

      :param dae:
      :type dae:
      :param i_d:
      :type i_d:
      :param i_q:
      :type i_q:
      :return:
      :rtype:



   .. py:method:: 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

      :param dae: an instance of a class Dae

      :type dae:

      :return: None

      :rtype: None



.. py:class:: SynchronousSubtransientSP

   Bases: :py:obj:`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**


   .. math::

       \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)

   **Electromagnetic Equations**


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

   .. math::

       \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)



   **Flux Linkage Dynamics**


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

   .. math::

       \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)

   **Algebraic Equations**

   The following algebraic equations govern the system:

   .. math::

       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}


   .. py:attribute:: _type
      :value: 'Synchronous_machine'



   .. py:attribute:: _name
      :value: 'Synchronous_machine_subtransient_model_Sauer_Pai'



   .. py:attribute:: e_dprim


   .. py:attribute:: e_qprim


   .. py:attribute:: psid


   .. py:attribute:: psiq


   .. py:attribute:: psid2


   .. py:attribute:: psiq2


   .. py:attribute:: x_l


   .. py:attribute:: gd1


   .. py:attribute:: gq1


   .. py:attribute:: gd2


   .. py:attribute:: gq2


   .. py:attribute:: x_dprim


   .. py:attribute:: x_qprim


   .. py:attribute:: T_dprim


   .. py:attribute:: T_qprim


   .. py:attribute:: x_dsec


   .. py:attribute:: x_qsec


   .. py:attribute:: T_dsec


   .. py:attribute:: T_qsec


   .. py:method:: 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



   .. py:method:: 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

      :param dae: an instance of a class Dae

      :type dae:

      :return: None

      :rtype: None



.. py:class:: SynchronousSubtransientSP6

   Bases: :py:obj:`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**


   .. math::

       \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)

   **Electromagnetic Equations**


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

   .. math::

       \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)



   **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:

   .. math::

       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}


   .. py:attribute:: _type
      :value: 'Synchronous_machine'



   .. py:attribute:: _name
      :value: 'Synchronous_machine_subtransient_model_Sauer_Pai_6th_order'



   .. py:attribute:: e_dprim


   .. py:attribute:: e_qprim


   .. py:attribute:: psid2


   .. py:attribute:: psiq2


   .. py:attribute:: x_l


   .. py:attribute:: gd1


   .. py:attribute:: gq1


   .. py:attribute:: gd2


   .. py:attribute:: gq2


   .. py:attribute:: x_dprim


   .. py:attribute:: x_qprim


   .. py:attribute:: T_dprim


   .. py:attribute:: T_qprim


   .. py:attribute:: x_dsec


   .. py:attribute:: x_qsec


   .. py:attribute:: T_dsec


   .. py:attribute:: T_qsec


   .. py:method:: 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 ():



   .. py:method:: 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

      :param dae: an instance of a class Dae

      :type dae:

      :return: None

      :rtype: None