ESST4B

Source: GridKit/Model/PhasorDynamics/Exciter/ESST4B/README.md

IEEE Type ST4B Potential- or Compound-Source Controlled-Rectifier Exciter Model (ESST4B)

ESST4B is a static excitation system with compensated-voltage sensing, an outer proportional/integral voltage regulator, a lag block, an inner proportional/integral regulator with exciter-output feedback, low-value over-excitation limiter gating, and potential- or compound-source rectifier scaling.

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 controlled-rectifier loading curve from Fig. 1; it is not a CommonMath helper.

• The potential-source calculation uses explicit real and imaginary terminal voltage/current components; the diagram’s complex expression is not used as model-equation notation below.

Block Diagram

Standard model of the ESST4B Exciter.

Figure 1: Exciter ESST4B model. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(T_R\)	[sec]	Tr	Compensated-voltage transducer time constant	0.0	Block name: Tr; if zero, sensed voltage is algebraic
\(K_{\mathrm{pr}}\)	[p.u.]	Kpr	Outer regulator proportional gain	1.0	Block name: KPR
\(K_{\mathrm{ir}}\)	[p.u./s]	Kir	Outer regulator integral gain	0.0	Block name: KIR
\(V_R^{\max}\)	[p.u.]	Vrmax	Maximum outer regulator output	1.0	Block name: VRMAX
\(V_R^{\min}\)	[p.u.]	Vrmin	Minimum outer regulator output	-1.0	Block name: VRMIN
\(T_A\)	[sec]	Ta	Regulator lag time constant	0.0	Block name: Ta; if zero, \(V_A\) is algebraic
\(K_{\mathrm{pm}}\)	[p.u.]	Kpm	Inner regulator proportional gain	1.0	Block name: KPM
\(K_{\mathrm{im}}\)	[p.u./s]	Kim	Inner regulator integral gain	0.0	Block name: KIM
\(V_M^{\max}\)	[p.u.]	VmMax	Maximum inner regulator output	1.0	Block name: VMMAX
\(V_M^{\min}\)	[p.u.]	VmMin	Minimum inner regulator output	0.0	Block name: VMMIN
\(K_G\)	[p.u.]	Kg	Exciter-output feedback gain into inner regulator	0.0	Block name: KG
\(K_P\)	[p.u.]	Kp	Potential-source voltage coefficient magnitude	0.0	Source label: KP
\(K_I\)	[p.u.]	Ki	Potential-source current coefficient	0.0	Source label: KI
\(V_B^{\max}\)	[p.u.]	VbMax	Maximum rectifier source multiplier	999.0	Block name: VBMAX
\(K_C\)	[p.u.]	Kc	Rectifier loading current coefficient	0.0	Block name: Kc; forms \(I_N\)
\(X_L\)	[p.u.]	Xl	Source reactance term in potential-source calculation	0.0	Source label: XL
\(\theta_P\)	[deg]	ThetaPDeg	Potential-source coefficient angle	0.0	Source label: thetaP; forms \(K_P^{\mathrm{r}}\) and \(K_P^{\mathrm{i}}\)
\(V_G^{\max}\)	[p.u.]	VgMax	Maximum exciter-output feedback signal	999.0	Block name: VGMAX; ceiling on \(K_G E_{\mathrm{fd}}\)

Parameter Validation

Invalid ESST4B parameter sets are rejected by the following checks.

\[\begin{split}\begin{aligned} &T_R \ge 0,\quad T_A \ge 0 \\ &V_R^{\min} \le V_R^{\max},\quad V_M^{\min} \le V_M^{\max} \\ &V_B^{\max} > 0,\quad V_G^{\max} \ge 0 \end{aligned}\end{split}\]

Model Derived Parameters

The potential-source coefficient is resolved into real scalar components:

\[\begin{split}\begin{aligned} K_P^{\mathrm{r}} &= K_P\cos\theta_P \\ K_P^{\mathrm{i}} &= K_P\sin\theta_P \end{aligned}\end{split}\]

Here \(\theta_P\) is converted from degrees before evaluating the trigonometric functions.

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(V_M\)	[p.u.]	Inner regulator output	State 1 in Fig. 1
\(V_C\)	[p.u.]	Sensed compensated voltage	State 2 in Fig. 1; source label: Sensed Vt; algebraic when \(T_R=0\)
\(V_A\)	[p.u.]	Lagged outer-regulator output	State 3 in Fig. 1; algebraic when \(T_A=0\)
\(x_R\)	[p.u.]	Outer regulator integral state	State 4 in Fig. 1; source label: VR

Algebraic

Symbol	Units	Description	Note
\(e_V\)	[p.u.]	Voltage-error signal into outer regulator	Summing junction after sensed voltage
\(V_R\)	[p.u.]	Limited outer regulator output	Limited by \(V_R^{\min}\) and \(V_R^{\max}\)
\(V_G\)	[p.u.]	Limited exciter-output feedback signal	\(K_G E_{\mathrm{fd}}\) limited by \(V_G^{\max}\)
\(e_M\)	[p.u.]	Inner regulator error	\(V_A\) minus \(V_G\)
\(V_{\mathrm{lv}}\)	[p.u.]	Low-value gate output	Lesser of \(V_M\) and \(V_{\mathrm{oel}}\)
\(V_{\mathrm{src}}^{\mathrm{r}}\)	[p.u.]	Real component of the potential-source expression	From terminal voltage/current components
\(V_{\mathrm{src}}^{\mathrm{i}}\)	[p.u.]	Imaginary component of the potential-source expression	From terminal voltage/current components
\(V_E\)	[p.u.]	Potential- or compound-source voltage magnitude	Nonnegative source magnitude
\(I_N\)	[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 \(F_{\mathrm{ex}}=f(I_N)\)
\(V_B\)	[p.u.]	Rectifier source multiplier	Limited by \(V_B^{\max}\)
\(E_{\mathrm{fd}}\)	[p.u.]	Field-voltage output	Product of low-value gate and \(V_B\)

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(V_{\mathrm{comp}}\)	[p.u.]	Compensated voltage input	Source label: VCOMP
\(V_{\mathrm{ref}}\)	[p.u.]	Voltage-control reference	Source label: VREF
\(V_{\mathrm{uel}}\)	[p.u.]	Under-excitation limiter input	Source label: VUEL; optional, defaults to zero
\(V_S\)	[p.u.]	Stabilizer input signal	Source label: VS; optional, defaults to zero
\(V_{\mathrm{oel}}\)	[p.u.]	Over-excitation limiter input	Source label: VOEL; optional, defaults to a high value when omitted
\(V_{\mathrm{r}}\)	[p.u.]	Terminal-voltage real component	Source label: VT
\(V_{\mathrm{i}}\)	[p.u.]	Terminal-voltage imaginary component	Source label: VT
\(I_{\mathrm{r}}\)	[p.u.]	Terminal-current real component	Source label: IT
\(I_{\mathrm{i}}\)	[p.u.]	Terminal-current imaginary component	Source label: IT
\(I_{\mathrm{fd}}\)	[p.u.]	Machine field current	Source label: IFD

Model Equations

Differential Equations

\[\begin{split}\begin{aligned} 0 &= -T_R\dot V_C - V_C + V_{\mathrm{comp}} \\ 0 &= -\dot x_R + \text{antiwindup}\!\left( V_R, K_{\mathrm{ir}}e_V, V_R^{\min}, V_R^{\max} \right) \\ 0 &= -T_A\dot V_A - V_A + V_R \\ 0 &= -\dot V_M + \text{antiwindup}\!\left( V_M, K_{\mathrm{im}}e_M, V_M^{\min}, V_M^{\max} \right) \end{aligned}\end{split}\]

CommonMath defines the Anti-Windup target and smooth approximation.

Algebraic Equations

\[\begin{split}\begin{aligned} 0 &= -e_V + V_{\mathrm{ref}} + V_{\mathrm{uel}} + V_S - V_C \\ 0 &= -V_R + \text{clamp}(K_{\mathrm{pr}}e_V + x_R, V_R^{\min}, V_R^{\max}) \\ 0 &= -V_G + \text{min}(K_G E_{\mathrm{fd}}, V_G^{\max}) \\ 0 &= -e_M + V_A - V_G \\ 0 &= -V_{\mathrm{lv}} + \text{min}(V_M, V_{\mathrm{oel}}) \\ 0 &= -V_{\mathrm{src}}^{\mathrm{r}} + K_P V_{\mathrm{r}} - X_L K_P^{\mathrm{i}} I_{\mathrm{r}} - \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{i}} \\ 0 &= -V_{\mathrm{src}}^{\mathrm{i}} + K_P V_{\mathrm{i}} + \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{r}} - X_L K_P^{\mathrm{i}} I_{\mathrm{i}} \\ 0 &= -V_E^2 + \left(V_{\mathrm{src}}^{\mathrm{r}}\right)^2 + \left(V_{\mathrm{src}}^{\mathrm{i}}\right)^2 \\ 0 &= -V_E I_N + K_C I_{\mathrm{fd}} \\ 0 &= -F_{\mathrm{ex}} + f(I_N) \\ 0 &= -V_B + \text{min}(V_E F_{\mathrm{ex}}, V_B^{\max}) \\ 0 &= -E_{\mathrm{fd}} + V_{\mathrm{lv}}V_B \end{aligned}\end{split}\]

CommonMath defines helper targets for min and clamp. The rectifier loading function \(f(I_N)\) is the source curve shown in Fig. 1.

Initialization

For a standard unsaturated start, the machine initializes \(E_{\mathrm{fd},0}\) and \(I_{\mathrm{fd},0}\) first. ESST4B reads those values, sets all internal derivatives to zero, and evaluates:

\[\begin{split}\begin{aligned} V_{C,0} &= V_{\mathrm{comp},0} \\ V_{\mathrm{src},0}^{\mathrm{r}} &= K_P V_{\mathrm{r},0} - X_L K_P^{\mathrm{i}} I_{\mathrm{r},0} - \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{i},0} \\ V_{\mathrm{src},0}^{\mathrm{i}} &= K_P V_{\mathrm{i},0} + \left(K_I + X_L K_P^{\mathrm{r}}\right)I_{\mathrm{r},0} - X_L K_P^{\mathrm{i}} I_{\mathrm{i},0} \\ V_{E,0} &= \sqrt{ \left(V_{\mathrm{src},0}^{\mathrm{r}}\right)^2 + \left(V_{\mathrm{src},0}^{\mathrm{i}}\right)^2 } \\ 0 &= -V_{E,0}I_{N,0} + K_C I_{\mathrm{fd},0} \\ F_{\mathrm{ex},0} &= f(I_{N,0}) \\ V_{B,0} &= \text{min}(V_{E,0}F_{\mathrm{ex},0}, V_B^{\max}) \\ V_{\mathrm{lv},0} &= \dfrac{E_{\mathrm{fd},0}}{V_{B,0}} \\ V_{M,0} &= V_{\mathrm{lv},0} \\ V_{G,0} &= \text{min}(K_G E_{\mathrm{fd},0}, V_G^{\max}) \\ e_{M,0} &= 0 \\ V_{A,0} &= V_{G,0} \\ V_{R,0} &= V_{A,0} \\ x_{R,0} &= V_{R,0} \\ e_{V,0} &= 0 \\ V_{\mathrm{ref},0} &= V_{C,0} - V_{\mathrm{uel},0} - V_{S,0} \end{aligned}\end{split}\]

This closed-form start requires \(V_{E,0}\ne 0\), \(V_{B,0}\ne 0\), inactive \(V_R\), \(V_M\), \(V_G\), and \(V_B\) limits, and the low-value gate selecting \(V_M\). Starts with active low-value gate limiting or saturated PI states are outside these closed-form equations.

Model Outputs

Output	Units	Description	Note
efd	[p.u.]	Field-voltage output	\(E_{\mathrm{fd}}\)
vm	[p.u.]	Inner regulator output	\(V_M\)
vc	[p.u.]	Sensed compensated voltage	\(V_C\)
va	[p.u.]	Lagged outer-regulator output	\(V_A\)
vr	[p.u.]	Outer regulator output	\(V_R\)
vg	[p.u.]	Exciter-output feedback signal	\(V_G\)
vlv	[p.u.]	Low-value gate output	\(V_{\mathrm{lv}}\)
ve	[p.u.]	Potential-source voltage magnitude	\(V_E\)
vb	[p.u.]	Rectifier source multiplier	\(V_B\)
in	[p.u.]	Normalized exciter loading current	\(I_N\)
fex	[p.u.]	Rectifier loading factor	\(F_{\mathrm{ex}}\)