# EXPIC1 _Source: `GridKit/Model/PhasorDynamics/Exciter/EXPIC1/README.md`_ ## **Proportional/Integral Excitation System Model (EXPIC1)** EXPIC1 is a proportional/integral excitation system with terminal-voltage sensing, a PI regulator, cascaded regulator filters, stabilizing feedback, potential/current-source scaling, rectifier loading, exciter limits, saturation, and an exciter field-voltage state. 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. - If $K_P=0$ and $K_I=0$, the diagram sets $V_B=1$. - If $T_E=0$, the source diagram states $E_{\mathrm{fd}}=E_0$; the exciter field state becomes algebraic. ### Block Diagram Standard model of the EXPIC1 Exciter. ```{image} ../../../../../Figures/PhasorDynamics/EXPIC1_diagram.png :align: center ``` Figure 1: Exciter EXPIC1 model. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/) ### Model Parameters Symbol | Units | JSON | Description | Typical Value | Note ------------------------------------|----------|-----------|---------------------------------------------------------|---------------|------ $T_R$ | [sec] | `Tr` | Transducer time constant | 0.0 | Block name: `Tr`; if zero, $E_T$ is algebraic $K_A$ | [p.u.] | `Ka` | PI regulator gain | 1.0 | Block name: `Ka` $T_{A1}$ | [sec] | `Ta1` | PI regulator numerator time constant | 0.0 | Block name: `Ta1` $V_{R1}^{\max}$ | [p.u.] | `Vr1` | PI regulator upper output limit | 1.0 | Source label: `VR1` $V_{R2}^{\min}$ | [p.u.] | `Vr2` | PI regulator lower output limit | -1.0 | Source label: `VR2` $T_{A2}$ | [sec] | `Ta2` | First denominator time constant in regulator filter | 0.0 | Block name: `Ta2` $T_{A3}$ | [sec] | `Ta3` | Numerator time constant in regulator filter | 0.0 | Block name: `Ta3` $T_{A4}$ | [sec] | `Ta4` | Second denominator time constant in regulator filter | 0.0 | Block name: `Ta4` $V_R^{\max}$ | [p.u.] | `Vrmax` | Maximum regulator output before source multiplier | 1.0 | Block name: `Vrmax` $V_R^{\min}$ | [p.u.] | `Vrmin` | Minimum regulator output before source multiplier | -1.0 | Block name: `Vrmin` $K_F$ | [p.u.] | `Kf` | Stabilizing feedback gain | 0.0 | Block name: `Kf` $T_{F1}$ | [sec] | `Tf1` | First feedback denominator time constant | 0.0 | Block name: `Tf1` $T_{F2}$ | [sec] | `Tf2` | Second feedback denominator time constant | 0.0 | Block name: `Tf2` $E_{\mathrm{fd}}^{\max}$ | [p.u.] | `Efdmax` | Maximum exciter input limit | 5.0 | Block name: `EFDMAX` $E_{\mathrm{fd}}^{\min}$ | [p.u.] | `Efdmin` | Minimum exciter input limit | -5.0 | Block name: `EFDMIN` $K_E$ | [p.u.] | `Ke` | Exciter field-resistance line-slope margin | 0.1 | Block name: `Ke` $T_E$ | [sec] | `Te` | Exciter time constant | 0.5 | Block name: `Te`; if zero, $E_{\mathrm{fd}}=E_0$ $E_1$ | [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` $K_P$ | [p.u.] | `Kp` | Potential-source voltage coefficient | 0.0 | Source label: `KP`; forms $V_E$ $K_I$ | [p.u.] | `Ki` | Potential-source current coefficient | 0.0 | Source label: `KI`; forms $V_E$ $K_C$ | [p.u.] | `Kc` | Rectifier loading current coefficient | 0.0 | Block name: `Kc`; forms $I_N$ #### Parameter Validation Invalid EXPIC1 parameter sets are rejected by the following checks. ```{math} \begin{aligned} &T_R \ge 0,\quad T_{A1}\ge 0,\quad T_{A2}\ge 0,\quad T_{A3}\ge 0,\quad T_{A4}\ge 0 \\ &T_{F1}\ge 0,\quad T_{F2}\ge 0,\quad T_E\ge 0 \\ &V_{R2}^{\min}\le V_{R1}^{\max},\quad V_R^{\min}\le V_R^{\max},\quad E_{\mathrm{fd}}^{\min}\le E_{\mathrm{fd}}^{\max} \end{aligned} ``` The saturation points are either disabled together or define a valid positive two-point 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: ```{math} \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} ``` The source calculation uses explicit real and imaginary terminal voltage/current components: ```{math} \begin{aligned} V_{\mathrm{src}}^{\mathrm{r}} &= K_P V_{\mathrm{r}} - K_I I_{\mathrm{i}} \\ V_{\mathrm{src}}^{\mathrm{i}} &= K_P V_{\mathrm{i}} + K_I I_{\mathrm{r}} \end{aligned} ``` ### Model Variables #### Internal Variables ##### Differential Symbol | Units | Description | Note ------------------------------------|--------|---------------------------------------------------------|------ $E_{\mathrm{fd}}$ | [p.u.] | Field-voltage output state | State 1 in Fig. 1; algebraic when $T_E=0$ $E_T$ | [p.u.] | Sensed terminal voltage | State 2 in Fig. 1; source label: `Sensed Vt`; algebraic when $T_R=0$ $V_A$ | [p.u.] | PI regulator output | State 3 in Fig. 1 $x_{R1}$ | [p.u.] | First regulator filter state | State 4 in Fig. 1; source label: `VR1` $V_R$ | [p.u.] | Regulator output before source multiplier | State 5 in Fig. 1; source label: `VR` $V_{F1}$ | [p.u.] | First feedback filter state | State 6 in Fig. 1; source label: `VF1` $V_F$ | [p.u.] | Stabilizing feedback output | State 7 in Fig. 1; source label: `VF` ##### Algebraic Symbol | Units | Description | Note ------------------------------------|--------|---------------------------------------------------------|------ $e_V$ | [p.u.] | Voltage-error signal after feedback | Summing junction after $E_T$ $V_{\mathrm{src}}^{\mathrm{r}}$ | [p.u.] | Real component of the source expression | From terminal voltage/current components $V_{\mathrm{src}}^{\mathrm{i}}$ | [p.u.] | Imaginary component of the source expression | From terminal voltage/current components $V_{\mathrm{src}}$ | [p.u.] | Potential/current source magnitude | Nonnegative source magnitude $I_N$ | [p.u.] | Normalized exciter loading current | Source label: `IN`; satisfies $V_{\mathrm{src}}I_N=K_C I_{\mathrm{fd}}$ when source scaling is active $F_{\mathrm{ex}}$ | [p.u.] | Rectifier loading factor | Source label: `FEX`; source curve $F_{\mathrm{ex}}=f(I_N)$ $V_B$ | [p.u.] | Source multiplier after rectifier loading | Product of $V_{\mathrm{src}}$ and $F_{\mathrm{ex}}$, or 1 when $K_P=K_I=0$ $E_0$ | [p.u.] | Limited exciter input | Limited by $E_{\mathrm{fd}}^{\min}$ and $E_{\mathrm{fd}}^{\max}$ $S_E$ | [p.u.] | Saturation coefficient evaluated at $E_{\mathrm{fd}}$ | Uses derived saturation curve #### 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` $V_{\mathrm{uel}}$ | [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 zero $V_{\mathrm{r}}$ | [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 E_T - E_T + E_C \\ 0 &= -\dot V_A + \text{antiwindup}\!\left( V_A, K_A e_V, V_{R2}^{\min}, V_{R1}^{\max} \right) \\ 0 &= -T_{A2}\dot x_{R1} - x_{R1} + V_A \\ 0 &= -T_{A4}\dot V_R - V_R + x_{R1} + T_{A3}\dot x_{R1} \\ 0 &= -T_{F1}\dot V_{F1} - V_{F1} + V_R \\ 0 &= -T_{F2}\dot V_F - V_F + K_F\dot V_{F1} \\ 0 &= -T_E\dot E_{\mathrm{fd}} + E_0 - (K_E + S_E)E_{\mathrm{fd}} \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_{\mathrm{oel}} - E_T - V_F \\ 0 &= -V_{\mathrm{src}}^{\mathrm{r}} + K_P V_{\mathrm{r}} - K_I I_{\mathrm{i}} \\ 0 &= -V_{\mathrm{src}}^{\mathrm{i}} + K_P V_{\mathrm{i}} + K_I I_{\mathrm{r}} \\ 0 &= -V_{\mathrm{src}}^2 + \left(V_{\mathrm{src}}^{\mathrm{r}}\right)^2 + \left(V_{\mathrm{src}}^{\mathrm{i}}\right)^2 \\ 0 &= \begin{cases} -I_N & K_P=0\ \text{and}\ K_I=0 \\ -V_{\mathrm{src}}I_N + K_C I_{\mathrm{fd}} & \text{otherwise} \end{cases} \\ 0 &= -F_{\mathrm{ex}} + \begin{cases} 1 & K_P=0\ \text{and}\ K_I=0 \\ f(I_N) & \text{otherwise} \end{cases} \\ 0 &= -V_B + \begin{cases} 1 & K_P=0\ \text{and}\ K_I=0 \\ V_{\mathrm{src}}F_{\mathrm{ex}} & \text{otherwise} \end{cases} \\ 0 &= -E_0 + \text{clamp}(V_B V_R, E_{\mathrm{fd}}^{\min}, E_{\mathrm{fd}}^{\max}) \\ 0 &= -S_E + S_B\,q(E_{\mathrm{fd}} - S_A) \end{aligned} ``` CommonMath defines helper targets for [clamp](../../../common-math.md#derived-functions) and the primitive [quadratic ramp](../../../common-math.md#primitives) $q$. The rectifier loading function $f(I_N)$ is the source curve shown in Fig. 1. The $V_{\mathrm{src}}$ residual uses the nonnegative branch of the squared source-magnitude equation. ### Initialization For a standard unsaturated start, the machine initializes $E_{\mathrm{fd},0}$ and $I_{\mathrm{fd},0}$ first. EXPIC1 reads those values, sets all internal derivatives to zero, and evaluates: ```{math} \begin{aligned} E_{T,0} &= E_{C,0} \\ V_{\mathrm{src},0}^{\mathrm{r}} &= K_P V_{\mathrm{r},0} - K_I I_{\mathrm{i},0} \\ V_{\mathrm{src},0}^{\mathrm{i}} &= K_P V_{\mathrm{i},0} + K_I I_{\mathrm{r},0} \\ V_{\mathrm{src},0} &= \sqrt{ \left(V_{\mathrm{src},0}^{\mathrm{r}}\right)^2 + \left(V_{\mathrm{src},0}^{\mathrm{i}}\right)^2 } \\ 0 &= \begin{cases} -I_{N,0} & K_P=0\ \text{and}\ K_I=0 \\ -V_{\mathrm{src},0}I_{N,0} + K_C I_{\mathrm{fd},0} & \text{otherwise} \end{cases} \\ F_{\mathrm{ex},0} &= \begin{cases} 1 & K_P=0\ \text{and}\ K_I=0 \\ f(I_{N,0}) & \text{otherwise} \end{cases} \\ V_{B,0} &= \begin{cases} 1 & K_P=0\ \text{and}\ K_I=0 \\ V_{\mathrm{src},0}F_{\mathrm{ex},0} & \text{otherwise} \end{cases} \\ S_{E,0} &= S_B\,q(E_{\mathrm{fd},0} - S_A) \\ E_{0,0} &= (K_E + S_{E,0})E_{\mathrm{fd},0} \\ V_{R,0} &= \dfrac{E_{0,0}}{V_{B,0}} \\ x_{R1,0} &= V_{R,0} \\ V_{A,0} &= x_{R1,0} \\ V_{F1,0} &= V_{R,0} \\ V_{F,0} &= 0 \\ e_{V,0} &= \dfrac{V_{A,0}}{K_A} \\ V_{\mathrm{ref},0} &= e_{V,0} + E_{T,0} + V_{F,0} - V_{\mathrm{uel},0} - V_{S,0} - V_{\mathrm{oel},0} \end{aligned} ``` This closed-form start requires nonzero $K_A$ and $V_{B,0}$, inactive PI and exciter limits, and residual consistency with the source curve. When $K_P$ and $K_I$ are not both zero, it also requires $V_{\mathrm{src},0}\ne 0$. If $T_E=0$, the final exciter residual is algebraic and requires $E_{\mathrm{fd},0}=E_{0,0}$. Starts that bind the PI regulator, cascaded regulator, or exciter limits are outside these closed-form equations. ### Model Outputs Output | Units | Description | Note ----------------|--------|-------------------------------------|------ `efd` | [p.u.] | Field-voltage output | $E_{\mathrm{fd}}$ `et` | [p.u.] | Sensed terminal voltage | $E_T$ `va` | [p.u.] | PI regulator state | $V_A$ `vr1` | [p.u.] | First regulator filter state | $x_{R1}$ `vr` | [p.u.] | Regulator output | $V_R$ `vf1` | [p.u.] | First feedback filter state | $V_{F1}$ `vf` | [p.u.] | Stabilizing feedback output | $V_F$ `vb` | [p.u.] | Source multiplier | $V_B$ `in` | [p.u.] | Normalized exciter loading current | $I_N$ `fex` | [p.u.] | Rectifier loading factor | $F_{\mathrm{ex}}$ `se` | [p.u.] | Saturation coefficient | $S_E$