EXDC1

EXDC1 is a direct-current excitation-system model with a voltage transducer, input lead–lag compensation, a limited voltage regulator, exciter saturation, and stabilizing feedback.

Notes

• Internal voltage signals are on component base.

• The speed input is machine speed deviation, so the field-voltage multiplier is \(1 + \omega\).

Block Diagram

Figure 1: EXDC1 exciter model. Figure courtesy of the PowerWorld EXDC1 model reference.

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(T_R\)	[s]	TBD	Voltage transducer time constant	0.0
\(K_A\)	[p.u.]	TBD	Voltage-regulator gain	40.0
\(T_A\)	[s]	TBD	Voltage-regulator time constant	0.1
\(T_B\)	[s]	TBD	Input lead–lag denominator time constant	0.0
\(T_C\)	[s]	TBD	Input lead–lag numerator time constant	0.0
\(V_R^{\max}\)	[p.u.]	TBD	Maximum voltage-regulator output	1.0
\(V_R^{\min}\)	[p.u.]	TBD	Minimum voltage-regulator output	-1.0
\(K_E\)	[p.u.]	TBD	Exciter field resistance line slope margin	0.1
\(T_E\)	[s]	TBD	Exciter time constant	0.5
\(K_F\)	[p.u.]	TBD	Stabilizing feedback gain	0.05
\(T_{F1}\)	[s]	TBD	Stabilizing feedback time constant	0.7
\(E_1\)	[p.u.]	TBD	First saturation voltage point	2.8
\(S_E(E_1)\)	[p.u.]	TBD	Saturation coefficient at \(E_1\)	0.08
\(E_2\)	[p.u.]	TBD	Second saturation voltage point	3.7
\(S_E(E_2)\)	[p.u.]	TBD	Saturation coefficient at \(E_2\)	0.33

JSON parameter names are not yet specified.

Parameter Validation

A valid EXDC1 parameter set must satisfy the following conditions:

\[\begin{split}\begin{aligned} K_A &> 0 \\ T_R, T_B, T_C, T_{F1} &\ge 0 \\ T_A, T_E &> 0 \\ T_B &> 0 \quad\text{or}\quad T_B = T_C = 0 \\ V_R^{\min} &\le V_R^{\max} \end{aligned}\end{split}\]

All parameters must be finite.

The saturation points are either disabled together,

\[S_E(E_1) = S_E(E_2) = 0\]

or define a valid two-point scaled-quadratic fit:

\[\begin{split}\begin{aligned} E_1, E_2 &> 0 \\ S_E(E_1), S_E(E_2) &\ge 0 \\ (E_2 - E_1) \left[S_E(E_2) - S_E(E_1)\right] &> 0 \end{aligned}\end{split}\]

Model Derived Parameters

The scaled saturation contribution is

\[E S_E(E) = S_B q(E - S_A)\]

where \(q\) is the quadratic ramp. When saturation is disabled,

\[S_A = S_B = 0\]

When one saturation value is zero,

\[\begin{split}\begin{aligned} S_E(E_1) = 0 &: \quad S_A = E_1, \qquad S_B = \dfrac{E_2 S_E(E_2)}{(E_2 - E_1)^2} \\ S_E(E_2) = 0 &: \quad S_A = E_2, \qquad S_B = \dfrac{E_1 S_E(E_1)}{(E_1 - E_2)^2} \end{aligned}\end{split}\]

When both saturation values are positive,

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

Model Ports

Name	Port	Init	Description
ec	Input	Known	Compensated terminal-voltage magnitude
speed	Input	Known	Machine speed deviation
vref	Input	Unknown	Voltage-control reference
vs	Input	Known	Stabilizer input signal
vuel	Input	Known	Under-excitation limiter input
voel	Input	Known	Over-excitation limiter input
efd	Output	Known	Field-voltage output

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(V_C\)	[p.u.]	Filtered terminal-voltage magnitude	Algebraic when \(T_R = 0\)
\(x_{\mathrm{LL}}\)	[p.u.]	Input lead–lag denominator state	Algebraic when \(T_B = 0\)
\(V_R\)	[p.u.]	Voltage-regulator output
\(E_{\mathrm{fd}}'\)	[p.u.]	Field-voltage state	Before the speed multiplier
\(V_F\)	[p.u.]	Stabilizing feedback state	Algebraic when \(T_{F1} = 0\)

Algebraic

Symbol	Units	Description	Note
\(e_V\)	[p.u.]	Voltage-error summing output
\(V_B\)	[p.u.]	Input lead–lag output
\(s_e\)	[p.u.]	Scaled-quadratic saturation contribution
\(V_{\mathrm{FE}}\)	[p.u.]	Exciter feedback drive
\(E_{\mathrm{fd}}\)	[p.u.]	Field-voltage output

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(E_C\)	[p.u.]	Compensated terminal-voltage magnitude
\(\omega\)	[p.u.]	Machine speed deviation
\(V_{\mathrm{ref}}\)	[p.u.]	Voltage-control reference
\(V_S\)	[p.u.]	Stabilizer input signal
\(V_{\mathrm{uel}}\)	[p.u.]	Under-excitation limiter input
\(V_{\mathrm{oel}}\)	[p.u.]	Over-excitation limiter input

Model Equations

Internal Equations

Differential

\[\begin{split}\begin{aligned} 0 &= -T_R \dot{V}_C - V_C + E_C \\ 0 &= -T_B \dot{x}_{\mathrm{LL}} - x_{\mathrm{LL}} + e_V \\ 0 &= -T_A \dot{V}_R + \text{antiwindup} ( V_R, -V_R + K_A V_B; V_R^{\min}, V_R^{\max} ) \\ 0 &= -T_E \dot{E}_{\mathrm{fd}}' + V_R - V_{\mathrm{FE}} \\ 0 &= -T_{F1} \dot{V}_F - V_F + \dfrac{K_F}{T_E} (V_R - V_{\mathrm{FE}}) \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= -e_V + V_{\mathrm{ref}} + V_S + V_{\mathrm{uel}} + V_{\mathrm{oel}} - V_C - V_F \\ 0 &= \begin{cases} -V_B + e_V & T_B = T_C = 0 \\ -T_B (V_B - x_{\mathrm{LL}}) + T_C (e_V - x_{\mathrm{LL}}) & T_B > 0 \end{cases} \\ 0 &= -s_e + S_B q(E_{\mathrm{fd}}' - S_A) \\ 0 &= -V_{\mathrm{FE}} + K_E E_{\mathrm{fd}}' + s_e \\ 0 &= -E_{\mathrm{fd}} + (1 + \omega) E_{\mathrm{fd}}' \end{aligned}\end{split}\]

The limiter and saturation use the CommonMath antiwindup and quadratic ramp functions.

External Equations

None.

Initialization

Input Initialization

\[\begin{split}\begin{aligned} E_C &\leftarrow \text{compensated terminal-voltage magnitude} \\ E_{\mathrm{fd}} &\leftarrow \text{machine field voltage} \\ \omega &\leftarrow \text{machine speed deviation or }0 \\ V_S &\leftarrow \text{stabilizer signal or }0 \\ V_{\mathrm{uel}} &\leftarrow \text{under-excitation limiter input or }0 \\ V_{\mathrm{oel}} &\leftarrow \text{over-excitation limiter input or }0 \end{aligned}\end{split}\]

Internal Initialization

\[\begin{split}\begin{aligned} V_C &\leftarrow E_C \\ E_{\mathrm{fd}}' &\leftarrow \dfrac{E_{\mathrm{fd}}}{1 + \omega} \\ s_e &\leftarrow S_B q(E_{\mathrm{fd}}' - S_A) \\ V_{\mathrm{FE}} &\leftarrow K_E E_{\mathrm{fd}}' + s_e \\ V_R &\leftarrow V_{\mathrm{FE}} \\ V_B &\leftarrow \dfrac{V_R}{K_A} \\ V_F &\leftarrow 0 \\ e_V &\leftarrow V_B \\ x_{\mathrm{LL}} &\leftarrow e_V \\ \dot{V}_C, \dot{x}_{\mathrm{LL}}, \dot{V}_R, \dot{E}_{\mathrm{fd}}', \dot{V}_F &\leftarrow 0 \end{aligned}\end{split}\]

Initialization requires \(1 + \omega > 0\) and \(V_R^{\min} \le V_R \le V_R^{\max}\).

Output Initialization

\[V_{\mathrm{ref}} \leftarrow e_V + V_C + V_F - V_S - V_{\mathrm{uel}} - V_{\mathrm{oel}}\]

Monitors

TBD.