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 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

Figure 2: GENSAL. Figure courtesy of PowerWorld

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

\[\begin{split}\begin{aligned} G &= \dfrac{R_a}{R_a^2+(X_d'')^2} & B &= -\dfrac{X_d''}{R_a^2+(X_d'')^2}\\ S_A &= \dfrac{1.2\sqrt{S_{10}/S_{12}} +1}{\sqrt{S_{10}/S_{12}} +1} & S_B &= \dfrac{1.2\sqrt{S_{10}/S_{12}} -1}{\sqrt{S_{10}/S_{12}} -1} \\ X_{d1} &= X_d-X_d' & X_{q2} &= X_q-X_d'' \\ X_{d2} &= X_d'-X_\ell & X_{d3} &= (X_d'-X_d'')/X_{d2}^2 \\ X_{d4} &= (X_d'-X_d'')/X_{d2} & X_{d5} &= (X_d''-X_\ell)/X_{d2} \\ f_\mathrm{base} &= f_\mathrm{sys} & S_\mathrm{mach,VA} &= 10^6 S_\mathrm{mach} \end{aligned}\end{split}\]

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

\[\begin{split}\begin{aligned} \dot\delta &= \omega \cdot 2\pi f_\mathrm{base} \\ \dot\omega &= \dfrac{1}{2H}\left(\dfrac{P_m-D\omega}{1+\omega} - T_e\right)\\ \dot{E}'_q &= \dfrac{1}{T'_{d0}} \left( E_{fd}-E'_q-X_{d1} (I_d+X_{d3}(E'_q-\psi'_d-X_{d2}I_d)) -k_{sat} \right)\\ \dot{\psi}'_d &= \dfrac{1}{T''_{d0}}(E'_q-\psi'_d-X_{d2}I_d)\\ \dot{\psi}''_q &= \dfrac{1}{T''_{q0}}(-\psi''_q-X_{q2}I_q) \end{aligned}\end{split}\]

Algebraic Equations

\[\begin{split}\begin{aligned} 0 &= -\psi''_d + E'_qX_{d5}+\psi'_dX_{d4}\\ 0 &= -k_{sat} + S_B(E'_q-S_A)^2\sigma(E'_q-S_A)\\ 0 &= -V_d -\psi''_q(1+\omega)\\ 0 &= -V_q +\psi''_d(1+\omega)\\ 0 &= -T_e +(\psi''_d-I_dX_d'')I_q-(\psi''_q-I_qX_d'')I_d\\ 0 &= -I_d + I_r \sin(\delta) - I_i \cos(\delta) \\ 0 &= -I_q + I_r \cos(\delta) + I_i \sin(\delta) \\ 0 &= -I_r + G (V_d \sin(\delta) + V_q \cos(\delta) - V_r) - B (-V_d \cos(\delta) + V_q \sin(\delta) - V_i) \\ 0 &= -I_i + B (V_d \sin(\delta) + V_q \cos(\delta) - V_r) + G (-V_d \cos(\delta) + V_q \sin(\delta) - V_i) \end{aligned}\end{split}\]

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.

\[\begin{split}\begin{aligned} \omega &= 0 \\ \delta &= \text{arg}\left[V_r+jV_i+(R_a+jX_q)(I_r+jI_i)\right]\\ I_d &= I_r\sin(\delta)-I_i\cos(\delta)\\ I_q &= I_r\cos(\delta)+I_i\sin(\delta)\\ \psi''_q &= -X_{q2}I_q\\ V_d &= -\psi''_q\\ V_q &= V_r\cos(\delta)+V_i\sin(\delta)+X_d''I_d+R_aI_q\\ \psi''_d &= V_q\\ \psi'_d &= \psi''_d-(X_d''-X_\ell)I_d\\ E'_q &= \psi'_d+X_{d2}I_d\\ k_{sat} &= S_B(E'_q-S_A)^2\sigma(E'_q-S_A)\\ T_e &= (\psi''_d-I_dX_d'')I_q-(\psi''_q-I_qX_d'')I_d\\ P_m &= T_e\\ E_{fd} &= E'_q+X_{d1}(I_d+X_{d3}(E'_q-\psi'_d-X_{d2}I_d))+k_{sat} \end{aligned}\end{split}\]

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