IEEEST

Source: GridKit/Model/PhasorDynamics/Stabilizer/IEEEST/README.md

IEEE Stabilizer Model (IEEEST)

Standard IEEE power system stabilizer: 4th-order notch filter, two lead–lag blocks, washout, and output limiter.

Block Diagram

Figure 1: Stabilizer IEEEST model. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	Description	Typical Value
\(A_1\)	[s]	Notch denominator coefficient	1.013
\(A_2\)	[s²]	Notch denominator coefficient	0.013
\(A_3\)	[s]	Notch denominator coefficient	0.0
\(A_4\)	[s²]	Notch denominator coefficient	0.0
\(A_5\)	[s]	Notch numerator coefficient	1.013
\(A_6\)	[s²]	Notch numerator coefficient	0.113
\(T_1\)	[s]	Lead–lag 1 numerator time constant	0.0
\(T_2\)	[s]	Lead–lag 1 denominator time constant	0.02
\(T_3\)	[s]	Lead–lag 2 numerator time constant	0.0
\(T_4\)	[s]	Lead–lag 2 denominator time constant	0.0
\(T_5\)	[s]	Washout numerator time constant	1.65
\(T_6\)	[s]	Washout denominator time constant	1.65
\(K_s\)	[p.u.]	Stabilizer gain	3.0
\(L_s^{\min}\)	[p.u.]	Minimum stabilizer output limit	-0.1
\(L_s^{\max}\)	[p.u.]	Maximum stabilizer output limit	0.1

The IEEE 421.5 IEEEST also defines a cutout window (\(V_{cl}\), \(V_{cu}\)) and an input delay (\(T_{delay}\)). These parameters are accepted for input-format compatibility but are not modeled here.

Derived Parameters

\[\begin{split}\begin{aligned} a_0 &= 1 \\ a_1 &= A_1 + A_3 \\ a_2 &= A_2 + A_4 + A_1 A_3 \\ a_3 &= A_1 A_4 + A_2 A_3 \\ a_4 &= A_2 A_4 \end{aligned}\end{split}\]

Model Variables

Internal Variables

Differential

Symbol	Units	Description
\(x_1, x_2, x_3, x_4\)	[-]	Notch filter states
\(x_5\)	[-]	Lead–lag 1 state
\(x_6\)	[-]	Lead–lag 2 state
\(x_7\)	[-]	Washout state

Algebraic

Symbol	Units	Description
\(v_4\)	[p.u.]	Notch filter output
\(v_5\)	[p.u.]	Lead–lag 1 output
\(v_6\)	[p.u.]	Lead–lag 2 output
\(v_7\)	[p.u.]	Unlimited stabilizer signal
\(V_{ss}\)	[p.u.]	Limited stabilizer signal (model output)

External Variables

Algebraic

Symbol	Units	Description
\(u\)	[p.u.]	Stabilizer input signal

Model Equations

Differential Equations

\[\begin{split}\begin{aligned} 0 &= -\dot{x}_1 + x_2 \\ 0 &= -\dot{x}_2 + x_3 \\ 0 &= -\dot{x}_3 + x_4 \\ 0 &= -\dot{x}_4 - \dfrac{a_0}{a_4}x_1 - \dfrac{a_1}{a_4}x_2 - \dfrac{a_2}{a_4}x_3 - \dfrac{a_3}{a_4}x_4 + \dfrac{1}{a_4}u \\ 0 &= -T_2 \dot{x}_5 - x_5 + v_4 \\ 0 &= -T_4 \dot{x}_6 - x_6 + v_5 \\ 0 &= -T_6 \dot{x}_7 - x_7 + v_6 \end{aligned}\end{split}\]

Algebraic Equations

\[\begin{split}\begin{aligned} 0 &= -v_4 + x_1 + A_5 x_2 + A_6 x_3 \\ 0 &= -T_2(v_5 - x_5) + T_1(v_4 - x_5) \\ 0 &= -T_4(v_6 - x_6) + T_3(v_5 - x_6) \\ 0 &= -T_6 v_7 + K_s T_5(v_6 - x_7) \\ 0 &= -V_{ss} + \text{clamp}(v_7, L_s^{\min}, L_s^{\max}) \end{aligned}\end{split}\]

The output limiter uses GridKit’s smooth Clamp.

Initialization

All states and their derivatives initialize to zero. The stabilizer comes online at rest and produces signal only in response to deviations in the input \(u\).