diff --git a/GridKit/Model/PhasorDynamics/Exciter/EXPIC1/README.md b/GridKit/Model/PhasorDynamics/Exciter/EXPIC1/README.md new file mode 100644 index 000000000..1073c54ac --- /dev/null +++ b/GridKit/Model/PhasorDynamics/Exciter/EXPIC1/README.md @@ -0,0 +1,272 @@ +# **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. + +
+ + + 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](../../../../CommonMath.md#anti-windup-indicator) +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](../../../../CommonMath.md#derived-functions) +and the primitive [quadratic ramp](../../../../CommonMath.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 | $S_E$ diff --git a/GridKit/Model/PhasorDynamics/Exciter/README.md b/GridKit/Model/PhasorDynamics/Exciter/README.md index 81144066b..0b6c03977 100644 --- a/GridKit/Model/PhasorDynamics/Exciter/README.md +++ b/GridKit/Model/PhasorDynamics/Exciter/README.md @@ -14,4 +14,5 @@ device internal voltage. There are a few standard Exciter models - IEEE Type 1 Excitation Model (See [IEEET1](IEEET1/README.md)) - IEEE DC1 Excitation Model (See [EXDC1](EXDC1/README.md)) +- EXPIC1 Excitation Model (See [EXPIC1](EXPIC1/README.md)) - Simplified Excitation System Model (See [SEXS-PTI](SEXS-PTI/README.md)) diff --git a/docs/Figures/PhasorDynamics/EXPIC1_diagram.png b/docs/Figures/PhasorDynamics/EXPIC1_diagram.png new file mode 100644 index 000000000..4bb1bdf44 Binary files /dev/null and b/docs/Figures/PhasorDynamics/EXPIC1_diagram.png differ