Generator Models

Here we discuss the structure and models used to model generators in LITS.jl. Each generator is a data structure that is defined by the following components:

Machine: That defines the stator electro-magnetic dynamics.
Shaft: That describes the rotor electro-mechanical dynamics.
Automatic Voltage Regulator: Electromotive dynamics to model an AVR controller.
Power System Stabilizer: Control dynamics to define an stabilization signal for the AVR.
Prime Mover and Turbine Governor: Thermo-mechanical dynamics and associated controllers.

The following figure summarizes the components of a generator and which variables they share:

⠀

Each generator is defined in its own $dq$ reference frame. Let $\delta$ be the rotor angle of the generator. If $v_r + jv_i = v_h\angle \theta$ defines the voltage in the bus in the network reference frame $RI$ rotating at nominal frequency $\Omega_b$, then the following equations (both are equivalent) can be used to convert between reference frames:

\[\begin{align} \left[ \begin{array}{c} v_d \\ v_q \end{array} \right] &= \left[ \begin{array}{c} v_h \sin(\delta - \theta) \\ v_h \cos(\delta - \theta) \end{array} \right] \tag{0a} \\ \left[ \begin{array}{c} v_d \\ v_q \end{array} \right] &= \left[ \begin{array}{cc} \sin(\delta) & -\cos(\delta) \\ \cos(\delta) & \sin(\delta) \end{array} \right] \left[ \begin{array}{c} v_r \\ v_i \end{array} \right] \tag{0b} \end{align}\]

Models are based from Federico Milano's book: "Power System Modelling and Scripting" and Prabha Kundur's book: "Power System's Stability and Control"

Machines

The machine component describes the stator-rotor electromagnetic dynamics.

Classical Model (Zero Order)

This is the classical order model that does not have differential equations in its machine model ($\delta$ and $\omega$ are defined in the shaft):

\[\begin{align} \left[ \begin{array}{c} i_d \\ i_q \end{array} \right] &= \left[ \begin{array}{cc} r_a & -x_d' \\ x_d' & r_a \end{array} \right]^{-1} \left[ \begin{array}{c} -v_d \\ e_q' - v_q \end{array} \right] \tag{1a}\\ p_e \approx \tau_e &= (v_q + r_a i_q)i_q + (v_d + r_ai_d)i_d \tag{1b} \end{align}\]

One d- One q- Model (2th Order)

This model includes two transient emf with their respective differential equations:

\[\begin{align} \dot{e}_q' &= \frac{1}{T_{d0}'} \left[-e_q' + (x_d-x_d')i_d + v_f\right] \tag{2a}\\ \dot{e}_d' &= \frac{1}{T_{q0}'} \left[-e_d' + (x_q-x_q')i_q \right] \tag{2b}\\ \left[ \begin{array}{c} i_d \\ i_q \end{array} \right] &= \left[ \begin{array}{cc} r_a & -x_q' \\ x_d' & r_a \end{array} \right]^{-1} \left[ \begin{array}{c} e_d'-v_d \\ e_q' - v_q \end{array} \right] \tag{2c}\\ p_e \approx \tau_e &= (v_q + r_a i_q)i_q + (v_d + r_ai_d)i_d \tag{2d} \end{align}\]

Marconato Machine (6th Order)

The Marconato model defines 6 differential equations, two for stator fluxes and 4 for transient and subtransient emfs:

\[\begin{align} \dot{\psi}_d &= \Omega_b(r_ai_d + \omega \psi_q + v_d) \tag{3a} \\ \dot{\psi}_q &= \Omega_b(r_ai_q - \omega \psi_d + v_q) \tag{3b} \\ \dot{e}_q' &= \frac{1}{T_{d0}'} \left[-e_q' - (x_d-x_d'-\gamma_d)i_d + \left(1- \frac{T_{AA}}{T_{d0}'} \right) v_f\right] \tag{3c}\\ \dot{e}_d' &= \frac{1}{T_{q0}'} \left[-e_d' + (x_q-x_q'-\gamma_q)i_q \right] \tag{3d}\\ \dot{e}_q'' &= \frac{1}{T_{d0}''} \left[-e_q'' + e_q' - (x_d'-x_d''+\gamma_d)i_d + \frac{T_{AA}}{T_{d0}'}v_f \right] \tag{3e} \\ \dot{e}_d'' &= \frac{1}{T_{q0}''} \left[-e_d'' + e_d' + (x_q'-x_q''+\gamma_q)i_q \right] \tag{3f} \\ i_d &= \frac{1}{x_d''} (e_q'' - \psi_d) \tag{3g} \\ i_q &= \frac{1}{x_q''} (-e_d'' - \psi_q) \tag{3h} \\ \tau_e &= \psi_d i_q - \psi_q i_d \tag{3i} \end{align}\]

with

\[\begin{align*} \gamma_d &= \frac{T_{d0}'' x_d''}{T_{d0}' x_d'} (x_d - x_d') \\ \gamma_q &= \frac{T_{q0}'' x_q''}{T_{q0}' x_q'} (x_q - x_q') \end{align*}\]

Simplified Marconato Machine (4th Order)

This model neglects the derivative of stator fluxes ($\dot{\psi}_d$ and $\dot{\psi}_q$) and assume that the rotor speed stays close to 1 pu ($\omega\psi_{d}=\psi_{d}$ and $\omega\psi_{q}=\psi_{q}$) that allows to remove the stator fluxes variables from the Marconato model.

\[\begin{align} \dot{e}_q' &= \frac{1}{T_{d0}'} \left[-e_q' - (x_d-x_d'-\gamma_d)i_d + \left(1- \frac{T_{AA}}{T_{d0}'} \right) v_f\right] \tag{4a}\\ \dot{e}_d' &= \frac{1}{T_{q0}'} \left[-e_d' + (x_q-x_q'-\gamma_q)i_q \right] \tag{4b}\\ \dot{e}_q'' &= \frac{1}{T_{d0}''} \left[-e_q'' + e_q' - (x_d'-x_d''+\gamma_d)i_d + \frac{T_{AA}}{T_{d0}'}v_f \right] \tag{4c} \\ \dot{e}_d'' &= \frac{1}{T_{q0}''} \left[-e_d'' + e_d' + (x_q'-x_q''+\gamma_q)i_q \right] \tag{4d} \\ \left[ \begin{array}{c} i_d \\ i_q \end{array} \right] &= \left[ \begin{array}{cc} r_a & -x_q'' \\ x_d'' & r_a \end{array} \right]^{-1} \left[ \begin{array}{c} e_d''-v_d \\ e_q'' - v_q \end{array} \right] \tag{4e}\\ p_e \approx \tau_e &= (v_q + r_a i_q)i_q + (v_d + r_ai_d)i_d \tag{4f} \end{align}\]

with

\[\begin{align*} \gamma_d &= \frac{T_{d0}'' x_d''}{T_{d0}' x_d'} (x_d - x_d') \\ \gamma_q &= \frac{T_{q0}'' x_q''}{T_{q0}' x_q'} (x_q - x_q') \end{align*}\]

Anderson-Fouad Machine (6th Order)

The Anderson-Fouad model also defines 6 differential equations, two for stator fluxes and 4 for transient and subtransient emfs and is derived from the Marconato model by defining $\gamma_d \approx \gamma_q \approx T_{AA} \approx 0$:

\[\begin{align} \dot{\psi}_d &= \Omega_b(r_ai_d + \omega \psi_q + v_d) \tag{5a} \\ \dot{\psi}_q &= \Omega_b(r_ai_q - \omega \psi_d + v_q) \tag{5b} \\ \dot{e}_q' &= \frac{1}{T_{d0}'} \left[-e_q' - (x_d-x_d')i_d + v_f\right] \tag{5c}\\ \dot{e}_d' &= \frac{1}{T_{q0}'} \left[-e_d' + (x_q-x_q')i_q \right] \tag{5d}\\ \dot{e}_q'' &= \frac{1}{T_{d0}''} \left[-e_q'' + e_q' - (x_d'-x_d'')i_d \right] \tag{5e} \\ \dot{e}_d'' &= \frac{1}{T_{q0}''} \left[-e_d'' + e_d' + (x_q'-x_q'')i_q \right] \tag{5f} \\ i_d &= \frac{1}{x_d''} (e_q'' - \psi_d) \tag{5g} \\ i_q &= \frac{1}{x_q''} (-e_d'' - \psi_q) \tag{5h} \\ \tau_e &= \psi_d i_q - \psi_q i_d \tag{5i} \end{align}\]

Simplified Anderson-Fouad Machine (4th Order)

Similar to the Simplified Marconato Model, this model neglects the derivative of stator fluxes ($\dot{\psi}_d$ and $\dot{\psi}_q$) and assume that the rotor speed stays close to 1 pu ($\omega \psi_d = \psi_d$ and $\omega \psi_q = \psi_q$) that allows to remove the stator fluxes variables from the model:

\[\begin{align} \dot{e}_q' &= \frac{1}{T_{d0}'} \left[-e_q' - (x_d-x_d')i_d + v_f\right] \tag{6a}\\ \dot{e}_d' &= \frac{1}{T_{q0}'} \left[-e_d' + (x_q-x_q')i_q \right] \tag{6b}\\ \dot{e}_q'' &= \frac{1}{T_{d0}''} \left[-e_q'' + e_q' - (x_d'-x_d'')i_d \right] \tag{6c} \\ \dot{e}_d'' &= \frac{1}{T_{q0}''} \left[-e_d'' + e_d' + (x_q'-x_q'')i_q \right] \tag{6d} \\ \left[ \begin{array}{c} i_d \\ i_q \end{array} \right] &= \left[ \begin{array}{cc} r_a & -x_q'' \\ x_d'' & r_a \end{array} \right]^{-1} \left[ \begin{array}{c} e_d''-v_d \\ e_q'' - v_q \end{array} \right] \tag{6e}\\ p_e \approx \tau_e &= (v_q + r_a i_q)i_q + (v_d + r_ai_d)i_d \tag{6f} \end{align}\]

Rotor Fluxes Model (5th Order)

This models describes a machine model using 2 stator fluxes $\psi_d$ and $\psi_q$, the rotor field flux $\psi_f$, the d-axis damping $\psi_{1d}$ and only one q-axis damping circuit $\psi_{1q}$:

\[\begin{align} \dot{\psi}_d &= \Omega_b(r_a i_d + \omega \psi_q + v_d) \tag{7a} \\ \dot{\psi}_q &= \Omega_b(r_ai_q - \omega \psi_d + v_q) \tag{7b} \\ \dot{\psi}_f &= -r_f i_f + v_f \tag{7c}\\ \dot{\psi}_{1d} &= -r_{1d}i_{1d} \tag{7d}\\ \dot{\psi}_{1q} &= -r_{1q}i_{1q} \tag{7e} \\ \left[ \begin{array}{c} i_d \\ i_f \\ i_{1d} \end{array} \right] &= \left[ \begin{array}{ccc} -L_d & -L_{ad} & L_{ad} \\ -L_{ad} & L_{ff} & L_{f1d} \\ -L_{ad} & L_{f1d} & L_{1d} \end{array} \right]^{-1} \left[ \begin{array}{c} \psi_d \\ \psi_f \\ \psi_{1d} \end{array} \right] \tag{7f}\\ \left[ \begin{array}{c} i_q \\ i_{1q} \end{array} \right] &= \left[ \begin{array}{ccc} -L_q & L_{aq} \\ -L_{aq} & L_{1q} \end{array} \right]^{-1} \left[ \begin{array}{c} \psi_q \\ \psi_{1q} \end{array} \right] \tag{7g}\\ \tau_e &= \psi_d i_q - \psi_q i_d \tag{7h} \end{align}\]

Shafts

The shaft component defines the rotating mass of the synchronous generator.

Rotor Mass Shaft

This is the standard model, on which one single mass (typically the rotor) is used to model the entire inertia of the synchronous generator. Each generator's rotating frame use a reference frequency $\omega_s$, that typically is the synchronous one (i.e. $\omega_s = 1.0$). The model defines two differential equations for the rotor angle $\delta$ and the rotor speed $\omega$:

\[\begin{align} \dot{\delta} &= \Omega_b(\omega - \omega_s) \tag{8a} \\ \dot{\omega} &= \frac{1}{2H}(\tau_m - \tau_e - D(\omega-\omega_s)) \tag{8b} \end{align}\]

Five-Mass Shaft

This model describes model connecting a high-pressure (hp) steam turbine, intermediate-pressure (ip) steam turbine, low-pressure (lp) steam pressure, rotor and exciter (ex) connected in series (in that order) in the same shaft using a spring-mass model:

\[\begin{align} \dot{\delta} &= \Omega_b(\omega - \omega_s) \tag{9a} \\ \dot{\omega} &= \frac{1}{2H} \left[- \tau_e - D(\omega-\omega_s)) - D_{34} (\omega-\omega_{lp}) - D_{45}(\omega-\omega_{ex}) + K_{lp}(\delta_{lp-\delta}) +K_{ex}(\delta_{ex}-\delta) \right] \tag{9b} \\ \dot{\delta}_{hp} &= \Omega_b(\omega_{hp} - \omega_s) \tag{9c} \\ \dot{\omega}_{hp} &= \frac{1}{2H_{hp}} \left[ \tau_m - D_{hp}(\omega_{hp}-\omega_s) - D_{12}(\omega_{hp} - \omega_{ip}) + K_{hp}(\delta_{ip} - \delta_{hp}) \right] \tag{9d} \\ \dot{\delta}_{ip} &= \Omega_b(\omega_{ip} - \omega_s) \tag{9e} \\ \dot{\omega}_{ip} &= \frac{1}{2H_{ip}} \left[- D_{ip}(\omega_{ip}-\omega_s) - D_{12}(\omega_{ip} - \omega_{hp}) -D_{23}(\omega_{ip} - \omega_{lp} ) + K_{hp}(\delta_{hp} - \delta_{ip}) + K_{ip}(\delta_{lp}-\delta_{ip}) \right] \tag{9f} \\ \dot{\delta}_{lp} &= \Omega_b(\omega_{lp}-\omega_s) \tag{9g} \\ \dot{\omega}_{lp} &= \frac{1}{2H_{lp}} \left[ - D_{lp}(\omega_{lp}-\omega_s) - D_{23}(\omega_{lp} - \omega_{ip}) -D_{34}(\omega_{lp} - \omega ) + K_{ip}(\delta_{ip} - \delta_{lp}) + K_{lp}(\delta-\delta_{lp}) \right] \tag{9h} \\ \dot{\delta}_{ex} &= \Omega_b(\omega_{ex}-\omega_s) \tag{9i} \\ \dot{\omega}_{ex} &= \frac{1}{2H_{ex}} \left[ - D_{ex}(\omega_{ex}-\omega_s) - D_{45}(\omega_{ex} - \omega) + K_{ex}(\delta - \delta_{ex}) \right] \tag{9j} \end{align}\]

Automatic Voltage Regulators (AVR)

AVR are used to determine the voltage in the field winding $v_f$ in the model.

Fixed AVR

This is a simple model that set the field voltage to be equal to a desired constant value $v_f = v_{\text{fix}}$.

Simple AVR

This depicts the most basic AVR, on which the field voltage is an integrator over the difference of the measured voltage and a reference:

\[\begin{align} \dot{v}_f = K_v(v_{\text{ref}} - v_h) \tag{10a} \end{align}\]

AVR Type I

This AVR is a simplified version of the IEEE DC1 AVR model:

\[\begin{align} \dot{v}_f &= -\frac{1}{T_e} \left[ V_f(K_e + S_e(v_f))-v_{r1} \right] \tag{11a} \\ \dot{v}_{r1} &= \frac{1}{T_a} \left[ K_a\left(v_{\text{ref}} - v_m - v_{r2} - \frac{K_f}{T_f}v_f\right) - v_{r1} \right] \tag{11b} \\ \dot{v}_{r2} &= -\frac{1}{T_f} \left[ \frac{K_f}{T_f}v_f + v_{r2} \right] \tag{11c} \\ \dot{v}_m &= \frac{1}{T_r} (v_h - v_m) \tag{11d} \end{align}\]

with the ceiling function:

\[\begin{align*} S_e(v_f) = A_e \exp(B_e|v_f|) \end{align*}\]

AVR Type II

This model represents a static exciter with higher gains and faster response than the Type I:

\[\begin{align} \dot{v}_f &= -\frac{1}{T_e} \left[ V_f(1 + S_e(v_f))-v_{r} \right] \tag{12a} \\ \dot{v}_{r1} &= \frac{1}{T_1} \left[ K_0\left(1 - \frac{T_2}{T_1} \right)(v_{\text{ref}} - v_m) - v_{r1} \right] \tag{12b} \\ \dot{v}_{r2} &= \frac{1}{K_0 T_3} \left[ \left( 1 - \frac{T_4}{T_3} \right) \left( v_{r1} + K_0\frac{T_2}{T_1}(v_{\text{ref}} - v_m)\right) - K_0 v_{r2} \right] \tag{12c} \\ \dot{v}_m &= \frac{1}{T_r} (v_h - v_m) \tag{12d} \end{align}\]

with

\[\begin{align*} v_r &= K_0v_{r2} + \frac{T_4}{T_3} \left( v_{r1} + K_0\frac{T_2}{T_1}(v_{\text{ref}} - v_m)\right) \\ S_e(v_f) &= A_e \exp(B_e|v_f|) \end{align*}\]

Power System Stabilizers (PSS)

PSS are used to add an additional signal $v_s$ to the field voltage: $v_f = v_f^{\text{avr}} + v_s$.

Fixed PSS

This is a simple model that set the stabilization signal to be equal to a desired constant value $v_s = v_{s}^{\text{fix}}$. The absence of PSS can be modelled using this component with $v_s^{\text{fix}} = 0$.

Simple PSS

This is the most basic PSS that can be implemented, on which the stabilization signal is simply a proportional controller over the frequency and electrical power:

\[\begin{align} v_s = K_{\omega}(\omega - \omega_s) + K_p(\omega \tau_e - P_{\text{ref}}) \tag{12a} \end{align}\]

Prime Movers and Turbine Governors (TG)

This section describes how mechanical power is modified to provide primary frequency control with synchronous generators. It is assumed that $\tau_{\text{ref}} = P_{\text{ref}}$ since they are decided at nominal frequency $\omega = 1$.

Fixed TG

This a simple model that set the mechanical torque to be equal to a proportion of the desired reference $\tau_m = \eta P_{\text{ref}}$. To set the mechanical torque to be equal to the desired power, the value of $\eta$ is set to 1.

TG Type I

This turbine governor is described by a droop controller and a low-pass filter to model the governor and two lead-lag blocks to model the servo and reheat of the turbine governor.

\[\begin{align} \dot{x}_{g1} &= \frac{1}{T_s}(p_{\text{in}} - x_{g1}) \tag{13a} \\ \dot{x}_{g2} &= \frac{1}{T_c} \left[ \left(1- \frac{t_3}{T_c}\right)x_{g1} - x_{g2} \right] \tag{13b} \\ \dot{x}_{g3} &= \frac{1}{T_5} \left[\left(1 - \frac{T_4}{T_5}\right)\left(x_{g2} + \frac{T_3}{T_c}x_{g1}\right) - x_{g3} \right] \tag{13c} \\ \tau_m &= x_{g3} + \frac{T_4}{T_5}\left(x_{g2} + \frac{T_3}{T_c}x_{g1}\right) \tag{13d} \end{align}\]

with

\[\begin{align*} p_{\text{in}} = P_{\text{ref}} + \frac{1}{R}(\omega_s - 1) \end{align*}\]

TG Type II

This turbine governor is a simplified model of the Type I.

\[\begin{align} \dot{x}_g &= \frac{1}{T_2}\left[\frac{1}{R}\left(1 - \frac{T_1}{T_2}\right) (\omega_s - \omega) - x_g\right] \tag{14a} \\ \tau_m &= P_{\text{ref}} + \frac{1}{R}\frac{T_1}{T_2}(\omega_s - \omega) \tag{14b} \end{align}\]

Reference

LITS.BaseMachine — Type.

Parameters of 0-states synchronous machine: Classical Model

Conmutable structor

BaseMachine(R, Xd_p, eq_p, MVABase)

Arguments

R::Float64 : Resistance after EMF in machine per unit
Xd_p::Float64 : Reactance after EMF in machine per unit
eq_p::Float64 : Fixed EMF behind the impedance
MVABase::Float64 : Nominal Capacity in MVA

LITS.OneDOneQMachine — Type.

Parameters of 2-states synchronous machine: One d- and One q-Axis Model

Conmutable structor

OneDOneQMachine(R, Xd, Xq, Xd_p, Xq_pp, Td0_p, Tq0_pp, MVABase)

Arguments

R::Float64
Xd::Float64
Xq::Float64
Xd_p::Float64 : Reactance after EMF in machine per unit
Xq_pp::Float64
Td0_p::Float64 : Time constant of transient d-axis voltage
Tq0_pp::Float64 : Time constant of subtransient q-axis voltage
MVABase::Float64 : Nominal Capacity in MVA

LITS.MarconatoMachine — Type.

Parameters of 6-states synchronous machine: Marconato model

Conmutable structor

MarconatoMachine(R, Xd, Xq, Xd_p, Xq_p, Xd_pp, Xq_pp, Td0_p, Tq0_p, Td0_pp, Tq0_pp, T_AA, MVABase)

Arguments

R::Float64 : Resistance after EMF in machine per unit
Xd::Float64 : Reactance after EMF in d-axis per unit
Xq::Float64 : Reactance after EMF in q-axis per unit
Xd_p::Float64 : Transient reactance after EMF in d-axis per unit
Xq_p::Float64 : Transient reactance after EMF in q-axis per unit
Xd_pp::Float64 : Subtransient reactance after EMF in d-axis per unit
Xq_pp::Float64 : Subtransient reactance after EMF in q-axis per unit
Td0_p::Float64 : Time constant of transient d-axis voltage
Tq0_p::Float64 : Time constant of transient q-axis voltage
Td0_pp::Float64 : Time constant of subtransient d-axis voltage
Tq0_pp::Float64 : Time constant of subtransient q-axis voltage
T_AA::Float64 : Time constant of d-axis additional leakage
MVABase::Float64 : Nominal Capacity in MVA

LITS.SimpleMarconatoMachine — Type.

Parameters of 4-states synchronous machine: Simplified Marconato model The derivative of stator fluxes (ψd and ψq) is neglected and ωψd = ψd and ωψq = ψq is assumed (i.e. ω=1.0). This is standard when transmission network dynamics is neglected.

Conmutable structor

SimpleMarconatoMachine(R, Xd, Xq, Xd_p, Xq_p, Xd_pp, Xq_pp, Td0_p, Tq0_p, Td0_pp, Tq0_pp, T_AA, MVABase)

Arguments

R::Float64 : Resistance after EMF in machine per unit
Xd::Float64 : Reactance after EMF in d-axis per unit
Xq::Float64 : Reactance after EMF in q-axis per unit
Xd_p::Float64 : Transient reactance after EMF in d-axis per unit
Xq_p::Float64 : Transient reactance after EMF in q-axis per unit
Xd_pp::Float64 : Subtransient reactance after EMF in d-axis per unit
Xq_pp::Float64 : Subtransient reactance after EMF in q-axis per unit
Td0_p::Float64 : Time constant of transient d-axis voltage
Tq0_p::Float64 : Time constant of transient q-axis voltage
Td0_pp::Float64 : Time constant of subtransient d-axis voltage
Tq0_pp::Float64 : Time constant of subtransient q-axis voltage
T_AA::Float64 : Time constant of d-axis additional leakage
MVABase::Float64 : Nominal Capacity in MVA

LITS.AndersonFouadMachine — Type.

Parameters of 6-states synchronous machine: Anderson-Fouad model

Conmutable structor

AndersonFouadMachine(R, Xd, Xq, Xd_p, Xq_p, Xd_pp, Xq_pp, Td0_p, Tq0_p, Td0_pp, Tq0_pp, MVABase)

Arguments

R::Float64 : Resistance after EMF in machine per unit
Xd::Float64 : Reactance after EMF in d-axis per unit
Xq::Float64 : Reactance after EMF in q-axis per unit
Xd_p::Float64 : Transient reactance after EMF in d-axis per unit
Xq_p::Float64 : Transient reactance after EMF in q-axis per unit
Xd_pp::Float64 : Subtransient reactance after EMF in d-axis per unit
Xq_pp::Float64 : Subtransient reactance after EMF in q-axis per unit
Td0_p::Float64 : Time constant of transient d-axis voltage
Tq0_p::Float64 : Time constant of transient q-axis voltage
Td0_pp::Float64 : Time constant of subtransient d-axis voltage
Tq0_pp::Float64 : Time constant of subtransient q-axis voltage
MVABase::Float64 : Nominal Capacity in MVA

LITS.SimpleAFMachine — Type.

Parameters of 4-states simplified Anderson-Fouad (SimpleAFMachine) model. The derivative of stator fluxes (ψd and ψq) is neglected and ωψd = ψd and ωψq = ψq is assumed (i.e. ω=1.0). This is standard when transmission network dynamics is neglected. If transmission dynamics is considered use the full order Anderson Fouad model.

Conmutable structor

SimpleAFMachine(R, Xd, Xq, Xd_p, Xq_p, Xd_pp, Xq_pp, Td0_p, Tq0_p, Td0_pp, Tq0_pp, MVABase)

Arguments

R::Float64 : Resistance after EMF in machine per unit
Xd::Float64 : Reactance after EMF in d-axis per unit
Xq::Float64 : Reactance after EMF in q-axis per unit
Xd_p::Float64 : Transient reactance after EMF in d-axis per unit
Xq_p::Float64 : Transient reactance after EMF in q-axis per unit
Xd_pp::Float64 : Subtransient reactance after EMF in d-axis per unit
Xq_pp::Float64 : Subtransient reactance after EMF in q-axis per unit
Td0_p::Float64 : Time constant of transient d-axis voltage
Tq0_p::Float64 : Time constant of transient q-axis voltage
Td0_pp::Float64 : Time constant of subtransient d-axis voltage
Tq0_pp::Float64 : Time constant of subtransient q-axis voltage
MVABase::Float64 : Nominal Capacity in MVA

LITS.FullMachine — Type.

Parameter of a full order flux stator-rotor model without zero sequence flux in the stator. The derivative of stator fluxes (ψd and ψq) is NOT neglected. Only one q-axis damping circuit is considered. All parameters are in machine per unit.

Refer to Chapter 3 of Power System Stability and Control by P. Kundur or Chapter 11 of Power System Dynamics: Stability and Control, by J. Machowski, J. Bialek and J. Bumby, for more details. Note that the models are somewhat different (but equivalent) due to the different Park Transformation used in both books.

Conmutable structor

FullMachine(R, R_f, R_1d, R_1q, L_d, L_q, L_ad, L_aq, L_f1d, L_f, L_1d, L_1q)

Arguments

R::Float64 : Stator resistance after EMF in per unit
R_f::Float64 : Field rotor winding resistance in per unit
R_1d::Float64 : Damping rotor winding resistance on d-axis in per unit. This value is denoted as RD in Machowski.
R_1q::Float64 : Damping rotor winding resistance on q-axis in per unit. This value is denoted as RQ in Machowski.
L_d::Float64 : Inductance of fictitious damping that represent the effect of the three-phase stator winding in the d-axis of the rotor, in per unit. This value is denoted as Lad + Ll in Kundur (and Ld in Machowski).
L_q::Float64 : Inductance of fictitious damping that represent the effect of the three-phase stator winding in the q-axis of the rotor, in per unit. This value is denoted as Laq + Ll in Kundur.
L_ad::Float64 : Mutual inductance between stator winding and rotor field (and damping) winding inductance on d-axis, in per unit
L_aq::Float64 : Mutual inductance between stator winding and rotor damping winding inductance on q-axis, in per unit
L_f1d::Float64 : Mutual inductance between rotor field winding and rotor damping winding inductance on d-axis, in per unit
L_ff::Float64 : Field rotor winding inductance, in per unit
L_1d::Float64 : Inductance of the d-axis rotor damping circuit, in per unit
L_1q::Float64 : Inductance of the q-axis rotor damping circuit, in per unit
MVABase::Float64 : Nominal Capacity in MVA

LITS.SingleMass — Type.

Parameters of single mass shaft model. Typically represents the rotor mass.

Conmutable structor

SingleMass(H, D)

Arguments

H ::Float64 : Rotor inertia constant in MWs/MVA
D ::Float64 : Rotor natural damping in pu

LITS.FiveMassShaft — Type.

Parameters of 5 mass-spring shaft model. It contains a High-Pressure (HP) steam turbine, Intermediate-Pressure (IP) steam turbine, Low-Pressure (LP) steam turbine, the Rotor and an Exciter (EX) mover.

Conmutable structor

FiveMassShaft(H, H_hp, H_ip, H_lp, H_ex,
              D, D_hp, D_ip, D_lp, D_ex,
              D_12, D_23, D_34, D_45,
              K_hp, K_ip, K_lp, K_ex)

Arguments

H ::Float64 : Rotor inertia constant in MWs/MVA
H_hp::Float64 : High pressure turbine inertia constant in MWs/MVA
H_ip::Float64 : Intermediate pressure turbine inertia constant in MWs/MVA
H_lp::Float64 : Low pressure turbine inertia constant in MWs/MVA
H_ex::Float64 : Exciter inertia constant in MWs/MVA
D ::Float64 : Rotor natural damping in pu
D_hp::Float64 : High pressure turbine natural damping in pu
D_ip::Float64 : Intermediate pressure turbine natural damping in pu
D_lp::Float64 : Low pressure turbine natural damping in pu
D_ex::Float64 : Exciter natural damping in pu
D_12::Float64 : High-Intermediate pressure turbine damping
D_23::Float64 : Intermediate-Low pressure turbine damping
D_34::Float64 : Low pressure turbine-Rotor damping
D_45::Float64 : Rotor-Exciter damping
K_hp::Float64 : High pressure turbine angle coefficient
K_ip::Float64 : Intermediate pressure turbine angle coefficient
K_lp::Float64 : Low pressure turbine angle coefficient
K_ex::Float64 : Exciter angle coefficient

LITS.AVRFixed — Type.

Parameters of a AVR that returns a fixed voltage to the rotor winding:

Conmutable structor

AVRFixed(Emf)

Arguments

Emf ::Float64 : Fixed voltage to the rotor winding

LITS.AVRSimple — Type.

Parameters of a simple proportional AVR in the derivative of EMF i.e. an integrator controller on EMF

Conmutable structor

AVRSimple(Kv)

#Arguments

Kv ::Float64 : Integral Gain

LITS.AVRTypeI — Type.

Parameters of an Automatic Voltage Regulator Type I - Resembles IEEE Type DC1

Conmutable structor

AVRTypeI(Ka, Ke, Kf, Ta, Tf, Te, Tr, Vr_max, Vr_min, Ae, Be)

Arguments

Ka::Float64 : Amplifier Gain
Ke::Float64 : Field circuit integral deviation
Kf::Float64 : Stabilizer Gain in s * pu/pu
Ta::Float64 : Amplifier Time Constant in s
Te::Float64 : Field Circuit Time Constant in s
Tf::Float64 : Stabilizer Time Constant in s
Tr::Float64 : Voltage Measurement Time Constant in s
Vr_max::Float64 : Maximum regulator voltage in pu
Vr_min::Float64 : Minimum regulator voltage in pu
Ae::Float64 : 1st ceiling coefficient
Be::Float64 : 2nd ceiling coefficient

LITS.AVRTypeII — Type.

Parameters of an Automatic Voltage Regulator Type II - Typical static exciter model

Conmutable structor

AVRTypeII(K0, T1, T2, T3, T4, Te, Tr, Vr_max, Vr_min, Ae, Be)

Arguments

K0::Float64 : Regulator Gain
T1::Float64 : First Pole in s
T2::Float64 : First zero in s
T3::Float64 : First Pole in s
T4::Float64 : First zero in s
Te::Float64 : Field Circuit Time Constant in s
Tr::Float64 : Voltage Measurement Time Constant in s
Vr_max::Float64 : Maximum regulator voltage in pu
Vr_min::Float64 : Minimum regulator voltage in pu
Ae::Float64 : 1st ceiling coefficient
Be::Float64 : 2nd ceiling coefficient

LITS.PSSFixed — Type.

Parameters of a PSS that returns a fixed voltage to add to the reference for the AVR

Conmutable structor

PSSFixed(Vs)

Arguments

Vs::Float64 : Fixed voltage stabilization signal

LITS.PSSSimple — Type.

Parameters of a PSS that returns a proportional droop voltage to add to the reference for the AVR

Conmutable structor

PSSSimple(K_ω, K_p)

Arguments

K_ω::Float64 : Proportional gain for frequency
K_p::Float64 : Proportional gain for active power

LITS.TGFixed — Type.

Parameters of a fixed Turbine Governor that returns a fixed mechanical torque given by the product of P_ref*efficiency

Conmutable structor

TGFixed(efficiency)

Arguments

efficiency::Float64 : Efficiency factor that multiplies P_ref

LITS.TGTypeI — Type.

Parameters of a Turbine Governor Type I.

Conmutable structor

TGTypeI(R, Ts, Tc, T3, T4, T5, P_min, P_max)

Arguments

R::Float64 : Droop parameter
Ts::Float64 : Governor time constant
Tc::Float64 : Servo time constant
T3::Float64 : Transient gain time constant
T4::Float64 : Power fraction time constant
T5::Float64 : Reheat time constant
P_min::Float64 : Min Power into the Governor
P_max::Float64 : Max Power into the Governor

LITS.TGTypeII — Type.

Parameters of a Turbine Governor Type II.

Conmutable structor

TGTypeI(R, T1, T2, τ_min, τ_max)

Arguments

R::Float64 : Droop parameter
T1::Float64 : Transient gain time constant
T2::Float64 : Governor time constant
τ_min::Float64 : Min turbine torque output
τ_max::Float64 : Max turbine torque output