Dynamic Models

Below is the list of all dynamic models used in the system. The corresponding classes are located in the ./devices directory. The following documentation describes the mathematical equations and the python implementation. Detailed instructions on how to create user-defined models aro not yet available.

Metaclass

All devices are derived from

class pydynamicestimator.devices.device.DeviceRect

Dynamic or static device modeled in rectangular coordinates. Used as a parent class for all devices modeled in rectangular coordinates.

properties

Flags for method calls (e.g., ‘gcall’, ‘fcall’).

Type:

dict[str, bool]

xf

Final state results for simulation/estimation.

Type:

dict[str, np.ndarray]

xinit

Initial state values for all devices of the instance.

Type:

dict[State, list[float]]

xmin

Minimum limited state value.

Type:

list[float]

xmax

Maximum limited state value.

Type:

list[float]

_params

Device parameters, such as rated voltage, power, and frequency and internal parameters.

Type:

dict[str, float]

_data

Additional data which will be used in a list, one entry for each device of this object instance.

Type:

dict[str, str]

bus

Buses where the device is connected. Each device has an entry in the list.

Type:

list[Optional[str]]

states

State variables.

Type:

list[States]

units

Units of state variables.

Type:

list[States]

ns

Total number of states in the model.

Type:

float

_states_noise

Noise for each state variable.

Type:

dict[States, float]

_states_init_error

Initial error for each state variable.

Type:

dict[States, float]

vre

Order of the real voltage value for each device in the overall DAE model.

Type:

list[float]

vim

Order of the real voltage value for each device in the overall DAE model.

Type:

list[float]

_algebs

Algebraic variables (‘vre’, ‘vim’).

Type:

list[str]

_descr

Descriptions for key parameters.

Type:

dict[str, str]

The method for generating the differential and algebraic equations in each model is:

pydynamicestimator.devices.device.DeviceRect.fgcall(self, dae)

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 (Dae) – an instance of a class Dae

Returns:

None

Return type:

None

—

Synchronous Generator Transient Model

The Synchronous Generator Transient Model provides a two-axis dynamic approximation of a synchronous generator. It is based on the formulation provided in [1].

Python Implementation

class pydynamicestimator.devices.synchronous.SynchronousTransient

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

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

Synchronous Generator Subtransient Model

The Synchronous Generator Subransient Model provides a dynamic approximation of a synchronous generator based on Anderson-Fauad’s model. It is based on the formulation provided in [1].

Python Implementation

class pydynamicestimator.devices.synchronous.SynchronousSubtransient

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

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

Synchronous Generator Subtransient Model (with Stator Dynamics)

The Synchronous Generator Subtransient Model with Stator Dynamics provides a dynamic approximation of a synchronous generator, including stator dynamics, based on the formulation by Sauer and Pai. It extends the standard subtransient model by incorporating additional stator field dynamics, as described in [3].

Python Implementation

class pydynamicestimator.devices.synchronous.SynchronousSubtransientSP

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:

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

Synchronous Generator Subtransient Model (without Stator Dynamics)

The Synchronous Generator Subtransient Model without Stator Dynamics provides a dynamic approximation of a synchronous generator, based on the formulation by Sauer and Pai. The stator dynamics are neglected making the model 6th order [3].

Python Implementation

class pydynamicestimator.devices.synchronous.SynchronousSubtransientSP6

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:

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

Converter Metaclass

This is the parent class for all inverter models. It introduces all necessary parameters and also stores the model initialization details with a custom script.

class pydynamicestimator.devices.inverter.Inverter

Metaclass for inverters

filter_init(dae)

Steady-state initialization of the inverter filter. Based on methods used in https://github.com/NREL-Sienna/PowerSimulationsDynamics.jl

Parameters:

dae (pydynamicestimator.system.Dae) – DAE object

Returns:

Vswd (ndarray): d-component of the switching block voltage Vswq (ndarray): q-component of the switching block voltage ifd_ext (ndarray): d-component of the filter current in the network (external) dq-reference frame ifq_ext (ndarray): q-component of the filter current in the network (external) dq-reference frame Vfd_ext (ndarray): d-component of the filter voltage in the network (external) dq-reference frame Vfq_ext (ndarray): q-component of the filter voltage in the network (external) dq-reference frame itd_ext (ndarray): d-component of the terminal current in the network (external) dq-reference frame itq_ext (ndarray): q-component of the terminal current in the network (external) dq-reference frame

Return type:

Tuple

finit(dae)

Parameters:

dae (pydynamicestimator.system.Dae)

Returns:

None

frequency_estimator_init(Vfd_ext, Vfq_ext)

Steady-state initialization of the frequency estimator. Based on methods used in https://github.com/NREL-Sienna/PowerSimulationsDynamics.jl

Return type:

Tuple[ndarray, ndarray, ndarray]

Parameters:

• Vfd_ext (ndarray) – d-component of the filter voltage in the network (external) dq-reference frame

• Vfq_ext (ndarray) – q-component of the filter voltage in the network (external) dq-reference frame

Returns:

Vfq_pll (ndarray): q-component of the filter voltage in the PLL’s dq-reference frame epsilon (ndarray): integrator state of the PLL delta_pll (ndarray): angle difference between the dq-reference frame of the PLL and the network

Return type:

Tuple

inner_loop_init(dae, ifd_ext, ifq_ext, Vfd_ext, Vfq_ext, Vswd, Vswq)

Steady-state initialization of the inverter inner loop. Based on methods used in https://github.com/NREL-Sienna/PowerSimulationsDynamics.jl

Return type:

Tuple[ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray]

Parameters:

• dae (pydynamicestimator.system.Dae) – DAE object

• ifd_ext (ndarray)

• ifq_ext (ndarray)

• Vfd_ext (ndarray)

• Vfq_ext (ndarray)

• Vswd (ndarray)

• Vswq (ndarray)

Returns:

delta_c Vref xi_d xi_q gamma_d gamma_q Pc_tilde Qc_tilde

Return type:

Tuple

outer_loop_init(dae, Vfd_ext, Vfq_ext)

Steady-state initialization of the inverter outer loop. Based on methods used in https://github.com/NREL-Sienna/PowerSimulationsDynamics.jl

Return type:

Tuple[ndarray, ndarray]

Parameters:

• dae (pydynamicestimator.system.Dae) – DAE object

• Vfd_ext (ndarray) – d-component of the filter voltage in the network (external) dq-reference frame

• Vfq_ext (ndarray) – q-component of the filter voltage in the network (external) dq-reference frame

Returns:

Pref (ndarray): active power reference Qref (ndarray): reactive power reference

Return type:

Tuple

Grid-Following Converter

The Grid-Following Converter Model provides a dynamic representation of a grid-following inverter. It captures the small-signal stability behavior in low-inertia power systems, as described in [2].

Python Implementation

class pydynamicestimator.devices.inverter.GridFollowing

Grid-Following Inverter (with Droop)

Based on the grid-following inverter model in https://doi.org/10.1109/TPWRS.2021.3061434

The dynamic behavior of the grid-following converter is described by the following differential equations:

Converter Voltage Dynamics

\[\dot{v}_{fd_{ext}} = \frac{\omega_{b}}{c_{f}}(i_{fd_{ext}} - i_{td_{ext}}) + \omega_{net}\omega_{b}v_{fq_{ext}}\]

\[\dot{v}_{fq_{ext}} = \frac{\omega_{b}}{c_{f}}(i_{fq_{ext}} - i_{tq_{ext}}) - \omega_{net}\omega_{b}v_{fd_{ext}}\]

Converter Current Dynamics

\[\dot{i}_{fd_{ext}} = \frac{\omega_{b}}{l_{f}}(v_{swd} - v_{fd_{ext}}) - \frac{\omega_{b}r_{f}}{l_{f}}i_{fd_{ext}} + \omega_{net}\omega_{b}i_{fq_{ext}}\]

\[\dot{i}_{fq_{ext}} = \frac{\omega_{b}}{l_{f}}(v_{swq} - v_{fq_{ext}}) - \frac{\omega_{b}r_{f}}{l_{f}}i_{fq_{ext}} - \omega_{net}\omega_{b}i_{fd_{ext}}\]

Grid-Side Current Dynamics

\[\dot{i}_{td_{ext}} = \frac{\omega_b}{l_t}(v_{fd_{ext}} - v_{n_{re}}) - \frac{\omega_b r_t}{l_t}i_{td_{ext}} + \omega_{net} \omega_b i_{tq_{ext}}\]

\[\dot{i}_{tq_{ext}} = \frac{\omega_b}{l_t}(v_{fq_{ext}} - v_{n_{im}}) - \frac{\omega_b r_t}{l_t}i_{tq_{ext}} - \omega_{net} \omega_b i_{td_{ext}}\]

Phase-Locked Loop (PLL) Dynamics

\[\dot{\epsilon} = v_{fq_{pll}}\]

\[\delta\dot{\theta}_{pll} = \omega_{b}\delta\omega_{pll}\]

Power and Frequency Dynamics

\[\dot{\tilde{p}}_{c} = \omega_{f}(p_{c} - \tilde{p}_{c})\]

\[\delta\dot{\theta}_{c} = \omega_{b}\delta\omega_{c}\]

\[\dot{\tilde{q}}_{c} = \omega_{f}(q_{c} - \tilde{q}_{c})\]

Control Dynamics

\[\dot{\xi}_{d} = v_{fd^{*}} - v_{fd_{int}}\]

\[\dot{\xi}_{q} = v_{fq^{*}} - v_{fq_{int}}\]

\[\dot{\gamma}_{d} = i_{fd^{*}} - i_{fd_{int}}\]

\[\dot{\gamma}_{q} = i_{fq^{*}} - i_{fq_{int}}\]

Grid-Forming Converter

The Grid-Forming Converter Model provides a dynamic representation of a grid-forming inverter. It captures the small-signal stability behavior in low-inertia power systems, as described in [2].

Python Implementation

class pydynamicestimator.devices.inverter.GridForming

Grid-Forming Inverter (Droop-Based)

Based on the grid-forming inverter model in https://doi.org/10.1109/TPWRS.2021.3061434 The dynamic behavior of the grid-forming converter is described by the following differential equations:

Converter Voltage Dynamics

\[\dot{v}_{fd_{ext}} = \frac{\omega_{b}}{c_{f}}(i_{fd_{ext}} - i_{td_{ext}}) + \omega_{net}\omega_{b}v_{fq_{ext}}\]

\[\dot{v}_{fq_{ext}} = \frac{\omega_{b}}{c_{f}}(i_{fq_{ext}} - i_{tq_{ext}}) - \omega_{net}\omega_{b}v_{fd_{ext}}\]

Converter Current Dynamics

\[\dot{i}_{fd_{ext}} = \frac{\omega_{b}}{l_{f}}(v_{swd} - v_{fd_{ext}}) - \frac{\omega_{b}r_{f}}{l_{f}}i_{fd_{ext}} + \omega_{net}\omega_{b}i_{fq_{ext}}\]

\[\dot{i}_{fq_{ext}} = \frac{\omega_{b}}{l_{f}}(v_{swq} - v_{fq_{ext}}) - \frac{\omega_{b}r_{f}}{l_{f}}i_{fq_{ext}} - \omega_{net}\omega_{b}i_{fd_{ext}}\]

Grid-Side Current Dynamics

\[\dot{i}_{td_{ext}} = \frac{\omega_b}{l_t}(v_{fd_{ext}} - v_{n_{re}}) - \frac{\omega_b r_t}{l_t}i_{td_{ext}} + \omega_{net} \omega_b i_{tq_{ext}}\]

\[\dot{i}_{tq_{ext}} = \frac{\omega_b}{l_t}(v_{fq_{ext}} - v_{n_{im}}) - \frac{\omega_b r_t}{l_t}i_{tq_{ext}} - \omega_{net} \omega_b i_{td_{ext}}\]

Power and Frequency Dynamics

\[\dot{\tilde{p}}_{c} = \omega_{f}(p_{c} - \tilde{p}_{c})\]

\[\delta\dot{\theta}_{c} = \omega_{b}\delta\omega_{c}\]

\[\dot{\tilde{q}}_{c} = \omega_{f}(q_{c} - \tilde{q}_{c})\]

Control Dynamics

\[\dot{\xi}_{d} = v_{fd^{*}} - v_{fd_{int}}\]

\[\dot{\xi}_{q} = v_{fq^{*}} - v_{fq_{int}}\]

\[\dot{\gamma}_{d} = i_{fd^{*}} - i_{fd_{int}}\]

\[\dot{\gamma}_{q} = i_{fq^{*}} - i_{fq_{int}}\]

References

[1] (1,2)

Federico Milano, Power System Modelling and Scripting, Springer Berlin Heidelberg, 2010, Power Systems series. Available at: https://books.google.ch/books?id=MQu7IqoLrfYC.

[2] (1,2)

21. Markovic, O. Stanojev, P. Aristidou, E. Vrettos, D. Callaway, and G. Hug, Understanding Small-Signal Stability of Low-Inertia Systems, IEEE Transactions on Power Systems, vol. 36, no. 5, pp. 3997–4017, Sep. 2021, doi: 10.1109/TPWRS.2021.3061434.

[3] (1,2)

16. 23. Sauer and M. A. Pai, Power System Dynamics and Stability, The University of Illinois at Urbana-Champaign, 1406 W. Green St., Urbana, IL 61801.