# 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. ```{image} ../../../../../Figures/PhasorDynamics/ESST4B_diagram.png :align: center ``` Figure 1: Exciter ESST4B model. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/) ### 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. ```{math} \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} ``` #### Model Derived Parameters The potential-source coefficient is resolved into real scalar components: ```{math} \begin{aligned} K_P^{\mathrm{r}} &= K_P\cos\theta_P \\ K_P^{\mathrm{i}} &= K_P\sin\theta_P \end{aligned} ``` 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 ```{math} \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} ``` CommonMath defines the [Anti-Windup](../../../common-math.md#derived-functions) target and smooth approximation. #### Algebraic Equations ```{math} \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} ``` CommonMath defines helper targets for [min and clamp](../../../common-math.md#derived-functions). 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: ```{math} \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} ``` 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 | $F_{\mathrm{ex}}$