EXAC2

Source: GridKit/Model/PhasorDynamics/Exciter/EXAC2/README.md

IEEE Type AC2 Excitation System Model (EXAC2)

EXAC2 is an IEEE Type AC excitation system with a voltage transducer, lead-lag input compensation, limited voltage-amplifier state, low-value gate, voltage regulator limits, exciter alternator state, stabilizing feedback, rectifier loading, saturation, and optional speed multiplier.

Notes:

• Internal voltage and current signals are on model base unless otherwise stated.

• The rectifier loading block \(F_{\mathrm{ex}}=f(I_N)\) is the source AC-exciter loading curve from Fig. 1; it is not a CommonMath helper.

• The source diagram labels the optional multiplier input as Speed; GridKit uses machine speed deviation, so the enabled multiplier is \(1+\omega\).

Block Diagram

Standard model of the EXAC2 Exciter.

Figure 1: Exciter EXAC2 model. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(T_R\)	[sec]	Tr	Transducer time constant	0.0	Block name: Tr; if zero, \(V_C\) is algebraic
\(T_B\)	[sec]	Tb	Lag time constant for voltage-regulator input lead-lag	0.0	Block name: Tb
\(T_C\)	[sec]	Tc	Lead time constant for voltage-regulator input lead-lag	0.0	Block name: Tc
\(K_A\)	[p.u.]	Ka	Voltage-amplifier gain	40.0	Block name: Ka
\(T_A\)	[sec]	Ta	Voltage-amplifier time constant	0.1	Block name: Ta
\(V_A^{\max}\)	[p.u.]	VaMax	Maximum voltage-amplifier output	1.0	Block name: VAMAX
\(V_A^{\min}\)	[p.u.]	VaMin	Minimum voltage-amplifier output	-1.0	Block name: VAMIN
\(K_B\)	[p.u.]	Kb	Regulator pre-limit gain after low-value gate	1.0	Block name: KB
\(V_R^{\max}\)	[p.u.]	Vrmax	Maximum voltage-regulator output	1.0	Block name: Vrmax
\(V_R^{\min}\)	[p.u.]	Vrmin	Minimum voltage-regulator output	-1.0	Block name: Vrmin
\(T_E\)	[sec]	Te	Exciter alternator time constant	0.5	Block name: Te
\(K_L\)	[p.u.]	Kl	Field-current limiter feedback gain	0.0	Block name: KL
\(K_H\)	[p.u.]	Kh	Regulator feedback path gain	0.0	Block name: KH
\(K_F\)	[p.u.]	Kf	Stabilizing feedback gain	0.05	Block name: Kf
\(T_F\)	[sec]	Tf	Stabilizing feedback time constant	0.7	Block name: Tf; if zero, \(V_F\) is algebraic
\(K_C\)	[p.u.]	Kc	Rectifier loading current coefficient	0.0	Block name: Kc; forms \(I_N\)
\(K_D\)	[p.u.]	Kd	Demagnetizing factor feedback gain	0.0	Block name: Kd
\(K_E\)	[p.u.]	Ke	Exciter field-resistance line-slope margin	0.1	Block name: Ke
\(V_{\mathrm{lr}}\)	[p.u.]	VLr	Low-value gate lower reference	0.0	Source label: VLR
\(E_1\)	[p.u.]	E1	First saturation voltage point	2.8	Block name: E1
\(S_E(E_1)\)	[p.u.]	SE1	Saturation value at \(E_1\)	0.08	Block name: Se1
\(E_2\)	[p.u.]	E2	Second saturation voltage point	3.7	Block name: E2
\(S_E(E_2)\)	[p.u.]	SE2	Saturation value at \(E_2\)	0.33	Block name: Se2
\(s_{\mathrm{spd}}\)	[binary]	Spdmlt	Speed multiplier flag	0	Block name: Spdmlt; 1 enables the speed multiplier

Parameter Validation

Invalid EXAC2 parameter sets are rejected by the following checks.

\[\begin{split}\begin{aligned} &K_A > 0 \\ &T_R \ge 0,\quad T_A > 0,\quad T_B \ge 0,\quad T_C \ge 0,\quad T_E > 0,\quad T_F \ge 0 \\ &T_B > 0\quad\text{or}\quad(T_B = 0\ \text{and}\ T_C = 0) \\ &V_A^{\min} \le V_A^{\max},\quad V_R^{\min} \le V_R^{\max} \\ &s_{\mathrm{spd}} \in \{0,1\} \end{aligned}\end{split}\]

The saturation points follow the same two-point validation used by other exciter READMEs: both saturation values are zero, or both points define a valid positive quadratic fit.

Model Derived Parameters

The saturation curve is fitted from the two supplied saturation points. If both saturation factors are zero, use \(S_A=0\) and \(S_B=0\). Otherwise:

\[\begin{split}\begin{aligned} C &= \sqrt{\dfrac{S_E(E_2)}{S_E(E_1)}} \\ S_A &= \dfrac{C E_1 - E_2}{C - 1} \\ S_B &= \dfrac{S_E(E_1)}{(E_1 - S_A)^2} \end{aligned}\end{split}\]

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(V_E\)	[p.u.]	Exciter alternator voltage state before output multipliers	State 1 in Fig. 1; source label: VE
\(V_C\)	[p.u.]	Sensed compensated voltage	State 2 in Fig. 1; source label: Sensed Vt; algebraic when \(T_R=0\)
\(V_A\)	[p.u.]	Limited voltage-amplifier output	State 3 in Fig. 1; source label: VA
\(x_{\mathrm{ll}}\)	[p.u.]	Lead-lag block state	State 4 in Fig. 1; source label: VLL
\(V_F\)	[p.u.]	Stabilizing feedback washout output	State 5 in Fig. 1; source label: VF; algebraic when \(T_F=0\)

Algebraic

Symbol	Units	Description	Note
\(e_V\)	[p.u.]	Voltage-regulator input error before lead-lag block	Summing junction after sensed voltage
\(V_{\mathrm{ll}}\)	[p.u.]	Lead-lag output	Input to voltage amplifier
\(V_H\)	[p.u.]	Regulator feedback path signal	Block name: KH
\(V_L\)	[p.u.]	Field-current limiter low-value gate input	Block name: KL; lower reference \(V_{\mathrm{lr}}\)
\(V_{\mathrm{lv}}\)	[p.u.]	Low-value gate output	Lesser of amplifier path and limiter path
\(V_R\)	[p.u.]	Voltage-regulator output	Limited by \(V_R^{\min}\) and \(V_R^{\max}\)
\(S_E\)	[p.u.]	Saturation coefficient evaluated at \(V_E\)	Uses derived saturation curve
\(I_N\)	[p.u.]	Normalized exciter loading current	Source label: IN; satisfies \(V_E I_N=K_C I_{\mathrm{fd}}\)
\(F_{\mathrm{ex}}\)	[p.u.]	Rectifier loading factor	Source label: FEX; source curve \(F_{\mathrm{ex}}=f(I_N)\)
\(V_{\mathrm{fe}}\)	[p.u.]	Exciter feedback signal	Sum of saturation/resistance and \(K_D I_{\mathrm{fd}}\) paths
\(E_{\mathrm{fd}}\)	[p.u.]	Field-voltage output	Output after rectifier loading and optional speed multiplier

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(E_C\)	[p.u.]	Compensated terminal voltage magnitude	Source label: EC
\(V_{\mathrm{ref}}\)	[p.u.]	Voltage-control reference	Source label: VREF
\(V_S\)	[p.u.]	Stabilizer input signal	Source label: VS; optional, defaults to zero
\(V_{\mathrm{uel}}\)	[p.u.]	Under-excitation limiter input	Source label: VUEL; optional, defaults to zero
\(V_{\mathrm{oel}}\)	[p.u.]	Over-excitation limiter input	Source label: VOEL; optional, defaults to zero
\(I_{\mathrm{fd}}\)	[p.u.]	Machine field current	Source label: IFD
\(\omega\)	[p.u.]	Machine speed deviation	Source label: Speed; optional when \(s_{\mathrm{spd}}=0\)

Model Equations

Differential Equations

\[\begin{split}\begin{aligned} 0 &= -T_R\dot V_C - V_C + E_C \\ 0 &= -T_B\dot x_{\mathrm{ll}} - x_{\mathrm{ll}} + e_V \\ 0 &= -T_A\dot V_A + \text{antiwindup}\!\left( V_A, -V_A + K_A V_{\mathrm{ll}}, V_A^{\min}, V_A^{\max} \right) \\ 0 &= -T_E\dot V_E + V_R - V_{\mathrm{fe}} \\ 0 &= -T_F\dot V_F - V_F + K_F\dot V_E \end{aligned}\end{split}\]

CommonMath defines the Anti-Windup target and smooth approximation.

Algebraic Equations

\[\begin{split}\begin{aligned} 0 &= -e_V + V_{\mathrm{ref}} + V_S + V_{\mathrm{uel}} + V_{\mathrm{oel}} - V_C - V_F \\ 0 &= -T_B(V_{\mathrm{ll}} - x_{\mathrm{ll}}) + T_C(e_V - x_{\mathrm{ll}}) \\ 0 &= -V_H + K_H V_{\mathrm{fe}} \\ 0 &= -V_L + V_{\mathrm{lr}} + K_L V_{\mathrm{fe}} \\ 0 &= -V_{\mathrm{lv}} + \text{min}(V_A + V_H,\ V_L) \\ 0 &= -V_R + \text{clamp}(K_B V_{\mathrm{lv}}, V_R^{\min}, V_R^{\max}) \\ 0 &= -S_E + S_B\,q(V_E - S_A) \\ 0 &= -V_E I_N + K_C I_{\mathrm{fd}} \\ 0 &= -F_{\mathrm{ex}} + f(I_N) \\ 0 &= -V_{\mathrm{fe}} + (K_E + S_E)V_E + K_D I_{\mathrm{fd}} \\ 0 &= -E_{\mathrm{fd}} + \left(1+s_{\mathrm{spd}}\omega\right)F_{\mathrm{ex}}V_E \end{aligned}\end{split}\]

CommonMath defines helper targets for min and clamp and the primitive quadratic ramp \(q\). The rectifier loading function \(f(I_N)\) is the source curve shown in Fig. 1. When \(T_B=T_C=0\), the lead-lag block is bypassed so \(V_{\mathrm{ll}}=e_V\).

Initialization

The machine initializes \(E_{\mathrm{fd}}\) and \(I_{\mathrm{fd}}\) first. For a standard unsaturated start, EXAC2 reads those values, sets all internal derivatives to zero, and first solves the coupled rectifier-loading equations:

\[\begin{split}\begin{aligned} 0 &= -V_{E,0}I_{N,0} + K_C I_{\mathrm{fd},0} \\ 0 &= -F_{\mathrm{ex},0} + f(I_{N,0}) \\ 0 &= -E_{\mathrm{fd},0} + \left(1+s_{\mathrm{spd}}\omega_0\right)F_{\mathrm{ex},0}V_{E,0} \end{aligned}\end{split}\]

Then evaluate the feedback path:

\[\begin{split}\begin{aligned} S_{E,0} &= S_B\,q(V_{E,0} - S_A) \\ V_{\mathrm{fe},0} &= (K_E + S_{E,0})V_{E,0} + K_D I_{\mathrm{fd},0} \end{aligned}\end{split}\]

Then solve the low-value gate and voltage-regulator chain:

\[\begin{split}\begin{aligned} V_{R,0} &= V_{\mathrm{fe},0} \\ V_{H,0} &= K_H V_{\mathrm{fe},0} \\ V_{L,0} &= V_{\mathrm{lr}} + K_L V_{\mathrm{fe},0} \\ V_{\mathrm{lv},0} &= \dfrac{V_{R,0}}{K_B} \\ V_{A,0} &= V_{\mathrm{lv},0} - V_{H,0} \\ V_{\mathrm{ll},0} &= \dfrac{V_{A,0}}{K_A} \\ V_{C,0} &= E_{C,0} \\ V_{F,0} &= 0 \\ x_{\mathrm{ll},0} &= e_{V,0} = V_{\mathrm{ll},0} \\ V_{\mathrm{ref},0} &= e_{V,0} + V_{C,0} + V_{F,0} - V_{S,0} - V_{\mathrm{uel},0} - V_{\mathrm{oel},0} \end{aligned}\end{split}\]

This standard start requires \(1+s_{\mathrm{spd}}\omega_0\ne 0\), \(V_{E,0}\ne 0\), \(K_A\ne 0\), \(K_B\ne 0\), inactive \(V_A\) and \(V_R\) limits, and the low-value gate selecting the amplifier path. Starts with active low-value gate limiting or saturated regulator states are outside these closed-form equations.

Model Outputs

Output	Units	Description	Note
efd	[p.u.]	Field-voltage output	\(E_{\mathrm{fd}}\)
ve	[p.u.]	Exciter alternator voltage state	\(V_E\)
vc	[p.u.]	Sensed compensated voltage	\(V_C\)
va	[p.u.]	Voltage-amplifier state	\(V_A\)
vr	[p.u.]	Voltage-regulator output	\(V_R\)
vll	[p.u.]	Lead-lag output	\(V_{\mathrm{ll}}\)
vf	[p.u.]	Stabilizing feedback state	\(V_F\)
vlv	[p.u.]	Low-value gate output	\(V_{\mathrm{lv}}\)
in	[p.u.]	Normalized exciter loading current	\(I_N\)
fex	[p.u.]	Rectifier loading factor	\(F_{\mathrm{ex}}\)
se	[p.u.]	Saturation coefficient	\(S_E\)