# GENSALwS _Source: `GridKit/Model/PhasorDynamics/SynchronousMachine/GENSALwS/README.md`_ ## GENSAL This synchronous machine model is 5th order and is specifically designed for salient-pole machines. It is a standard model used in phasor-domain industry stability studies. See the [General Synchronous Machine Model](../index.md) for general synchronous machine information. Notes: - $X_q''=X_d''$ (no subtransient saliency) - $X_q=X_q'$ - $T'_{q0}$ is neglected - Only d-axis affected by saturation ### Block Diagram ```{image} ../../../../../Figures/GENSAL.JPG :align: center ``` Figure 2: GENSAL. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/) ### Model Parameters Symbol | Units | Description | Typical Value | Note ------------|---------|--------------------------------------------|---------------| ------ $P_0$ | [p.u.] | Initial active power injection | 1.0 | $Q_0$ | [p.u.] | Initial reactive power injection | 0.0 | $H$ | [s] | rotor inertia | 3 $D$ | [p.u.] | damping coefficient | 0 $R_a$ | [p.u.] | winding resistance | 0 $T'_{d0}$ | [s] | Open circuit direct axis transient time const. | 7 | $T''_{d0}$ | [s] | Open circuit direct axis sub-transient time const. | 0.04 | $T''_{q0}$ | [s] | Open circuit quadrature axis sub-transient time const. | 0.05 | $X_d$ | [p.u.] | Direct axis synchronous reactance | 2.1 | $X'_d$ | [p.u.] | Direct axis transient reactance | 0.2 | $X''_d$ | [p.u.] | Direct axis sub-transient reactance | 0.18 | $X_q$ | [p.u.] | Quadrature axis synchronous reactance | 0.5 | $X_{\ell}$ | [p.u.] | Stator leakage reactance | 0.15 | $S_{10}$ | [p.u.] | Saturation factor at 1.0 pu flux | 0 | $S_{12}$ | [p.u.] | Saturation factor at 1.2 pu flux | 0 | $S_\mathrm{mach}$ | [MVA] | Machine power base | 100 | #### Model Derived Parameters ```{math} \begin{aligned} G &= \dfrac{R_a}{R_a^2+(X_d'')^2} & B &= -\dfrac{X_d''}{R_a^2+(X_d'')^2}\\ S_A &= \dfrac{1.2\sqrt{S_{10}/S_{12}} +1}{\sqrt{S_{10}/S_{12}} +1} & S_B &= \dfrac{1.2\sqrt{S_{10}/S_{12}} -1}{\sqrt{S_{10}/S_{12}} -1} \\ X_{d1} &= X_d-X_d' & X_{q2} &= X_q-X_d'' \\ X_{d2} &= X_d'-X_\ell & X_{d3} &= (X_d'-X_d'')/X_{d2}^2 \\ X_{d4} &= (X_d'-X_d'')/X_{d2} & X_{d5} &= (X_d''-X_\ell)/X_{d2} \\ f_\mathrm{base} &= f_\mathrm{sys} & S_\mathrm{mach,VA} &= 10^6 S_\mathrm{mach} \end{aligned} ``` System bases are taken from the system at initialization. ### Model Variables #### Internal Variables ##### Differential Symbol | Units | Description | Note -------------|--------|-----------------------------------|------- $\delta$ | [rad] | Machine internal rotor angle | $\omega$ | [p.u.] | Machine speed deviation | Optionally read by governor or stabilizer component $E'_q$ | [p.u.] | Quadrature axis transient flux | $\psi'_d$ | [p.u.] | Direct axis transient flux | $\psi''_q$ | [p.u.] | Total q-axis subtransient flux | ##### Algebraic Symbol | Units | Description | Note ------------|--------|-----------------------------------| ------ $\psi''_d$ | [p.u.] | Total d-axis subtransient flux | $k_{sat}$ | [p.u.] | Additive saturation signal | $V_d$ | [p.u.] | Machine internal voltage, d-axis | $V_q$ | [p.u.] | Machine internal voltage, q-axis | $T_e$ | [p.u.] | Electrical torque | $I_d$ | [p.u.] | Terminal current, d-axis | $I_q$ | [p.u.] | Terminal current, q-axis | $I_r$ | [p.u.] | Terminal current, real component on network reference frame | Read by bus and optionally by controllers $I_i$ | [p.u.] | Terminal current, imaginary component on network reference frame | Read by bus and optionally by controllers #### External Variables ##### Differential None. ##### Algebraic Symbol | Units | Description | Note ---------|--------|---------------------------------------------------------| ------ $V_r$ | [p.u.] | Terminal voltage, real component on network reference frame | owned by bus object $V_i$ | [p.u.] | Terminal voltage, imaginary component on network reference frame | owned by bus object $P_m$ | [p.u.] | Mechanical power from the prime mover | Owned by governor, constant if no governor is connected to the machine $E_{fd}$ | [p.u.] | Field winding voltage from the excitation system | Owned by exciter, constant if no exciter is connected to the machine ### Model Equations #### Differential Equations ```{math} \begin{aligned} \dot\delta &= \omega \cdot 2\pi f_\mathrm{base} \\ \dot\omega &= \dfrac{1}{2H}\left(\dfrac{P_m-D\omega}{1+\omega} - T_e\right)\\ \dot{E}'_q &= \dfrac{1}{T'_{d0}} \left( E_{fd}-E'_q-X_{d1} (I_d+X_{d3}(E'_q-\psi'_d-X_{d2}I_d)) -k_{sat} \right)\\ \dot{\psi}'_d &= \dfrac{1}{T''_{d0}}(E'_q-\psi'_d-X_{d2}I_d)\\ \dot{\psi}''_q &= \dfrac{1}{T''_{q0}}(-\psi''_q-X_{q2}I_q) \end{aligned} ``` #### Algebraic Equations ```{math} \begin{aligned} 0 &= -\psi''_d + E'_qX_{d5}+\psi'_dX_{d4}\\ 0 &= -k_{sat} + S_B(E'_q-S_A)^2\sigma(E'_q-S_A)\\ 0 &= -V_d -\psi''_q(1+\omega)\\ 0 &= -V_q +\psi''_d(1+\omega)\\ 0 &= -T_e +(\psi''_d-I_dX_d'')I_q-(\psi''_q-I_qX_d'')I_d\\ 0 &= -I_d + I_r \sin(\delta) - I_i \cos(\delta) \\ 0 &= -I_q + I_r \cos(\delta) + I_i \sin(\delta) \\ 0 &= -I_r + G (V_d \sin(\delta) + V_q \cos(\delta) - V_r) - B (-V_d \cos(\delta) + V_q \sin(\delta) - V_i) \\ 0 &= -I_i + B (V_d \sin(\delta) + V_q \cos(\delta) - V_r) + G (-V_d \cos(\delta) + V_q \sin(\delta) - V_i) \end{aligned} ``` ### Initialization Using the power-flow solution, initial currents are calculated from active and reactive power injection. The remaining variables are initialized from the steady-state GENSAL equations. ```{math} \begin{aligned} \omega &= 0 \\ \delta &= \text{arg}\left[V_r+jV_i+(R_a+jX_q)(I_r+jI_i)\right]\\ I_d &= I_r\sin(\delta)-I_i\cos(\delta)\\ I_q &= I_r\cos(\delta)+I_i\sin(\delta)\\ \psi''_q &= -X_{q2}I_q\\ V_d &= -\psi''_q\\ V_q &= V_r\cos(\delta)+V_i\sin(\delta)+X_d''I_d+R_aI_q\\ \psi''_d &= V_q\\ \psi'_d &= \psi''_d-(X_d''-X_\ell)I_d\\ E'_q &= \psi'_d+X_{d2}I_d\\ k_{sat} &= S_B(E'_q-S_A)^2\sigma(E'_q-S_A)\\ T_e &= (\psi''_d-I_dX_d'')I_q-(\psi''_q-I_qX_d'')I_d\\ P_m &= T_e\\ E_{fd} &= E'_q+X_{d1}(I_d+X_{d3}(E'_q-\psi'_d-X_{d2}I_d))+k_{sat} \end{aligned} ``` ### Model Outputs Symbol | Units | Description | Note -----------|--------|-----------------------------------|------ $I_r$ | [p.u.] | Terminal current, real component on network reference frame | Oriented leaving the machine, system base $I_i$ | [p.u.] | Terminal current, imaginary component on network reference frame | Oriented leaving the machine, system base $P$ | [p.u.] | Active power, $V_rI_r+V_iI_i$ | Oriented leaving the machine, system base $Q$ | [p.u.] | Reactive power, $V_iI_r-V_rI_i$ | Oriented leaving the machine, system base $\delta$ | [rad] | Machine internal rotor angle | $\omega$ | [p.u.] | Machine speed deviation | $\omega=0$ at synchronous speed $\text{speed}$ | [p.u.] | Per-unit machine speed | $1+\omega$ $E'_q$ | [p.u.] | Quadrature axis transient flux | Machine base $\psi'_d$ | [p.u.] | Direct axis transient flux | Machine base $\psi''_q$ | [p.u.] | Total q-axis subtransient flux | Machine base $\psi''_d$ | [p.u.] | Total d-axis subtransient flux | Machine base $V_d$ | [p.u.] | Machine internal voltage, d-axis | Machine base $V_q$ | [p.u.] | Machine internal voltage, q-axis | Machine base $T_e$ | [p.u.] | Electrical torque | Machine base $I_d$ | [p.u.] | Terminal current, d-axis | Machine base $I_q$ | [p.u.] | Terminal current, q-axis | Machine base