IEEEST

Standard IEEE power system stabilizer: notch filter of order up to four, two lead–lag blocks, washout, and output limiter.

Block Diagram

Figure 1: Stabilizer IEEEST model. Figure courtesy of PowerWorld

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(A_1\)	[s]	A1	Notch denominator coefficient	1.013
\(A_2\)	[s²]	A2	Notch denominator coefficient	0.013
\(A_3\)	[s]	A3	Notch denominator coefficient	0.0
\(A_4\)	[s²]	A4	Notch denominator coefficient	0.0
\(A_5\)	[s]	A5	Notch numerator coefficient	1.013
\(A_6\)	[s²]	A6	Notch numerator coefficient	0.113
\(T_1\)	[s]	T1	Lead–lag 1 numerator time constant	0.0
\(T_2\)	[s]	T2	Lead–lag 1 denominator time constant	0.02
\(T_3\)	[s]	T3	Lead–lag 2 numerator time constant	0.0
\(T_4\)	[s]	T4	Lead–lag 2 denominator time constant	0.0
\(T_5\)	[s]	T5	Washout numerator time constant	1.65
\(T_6\)	[s]	T6	Washout denominator time constant	1.65
\(K_\mathrm{s}\)	[p.u.]	Ks	Stabilizer gain	3.0
\(L_\mathrm{s}^{\min}\)	[p.u.]	Lsmin	Minimum stabilizer output limit	-0.1
\(L_\mathrm{s}^{\max}\)	[p.u.]	Lsmax	Maximum stabilizer output limit	0.1

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

Parameter Validation

A valid IEEEST parameter set must satisfy the following conditions:

\[a_2=a_3=a_4=0 \quad\Longrightarrow\quad a_1=0\]

Model Derived Parameters

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

The notch order is

\[\begin{split}n = \begin{cases} 4 & a_4\ne0 \\ 3 & a_4=0,\ a_3\ne0 \\ 2 & a_4=a_3=0,\ a_2\ne0 \\ 0 & a_4=a_3=a_2=a_1=0 \end{cases}\end{split}\]

Model Ports

Name	Port	Init	Description
input	Input	Known	Required stabilizer input signal
output	Output	Known	Limited stabilizer output signal

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(x_1\)	[p.u.]	Notch filter state	Held at zero when \(n=0\)
\(x_2\)	[p.u./s]	Notch filter state	Held at zero when \(n=0\)
\(x_3\)	[p.u./s²]	Notch filter state	Held at zero when \(n<3\)
\(x_4\)	[p.u./s³]	Notch filter state	Held at zero when \(n<4\)
\(x_5\)	[p.u.]	Lead–lag 1 state	Algebraic when \(T_2=0\)
\(x_6\)	[p.u.]	Lead–lag 2 state	Algebraic when \(T_4=0\)
\(x_7\)	[p.u.]	Washout state	Algebraic when \(T_6=0\)

Algebraic

Symbol	Units	Description	Note
\(v_4\)	[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
\(V_{\mathrm{ss}}\)	[p.u.]	Limited stabilizer signal (model output)

External Variables

Differential

None.

Algebraic

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

Model Equations

Smooth functions: clamp.

Internal Equations

Differential

\[\begin{split}\begin{aligned} 0 &= -\dot{x}_k+x_{k+1}, && 1\le k<n \\ 0 &= -a_n\dot{x}_n-\sum_{j=0}^{n-1}a_jx_{j+1}+u, && n>0 \\ 0 &= -\dot{x}_k, && n<k\le4 \\ 0 &= -T_2\dot{x}_5-x_5+v_4 \\ 0 &= -T_4\dot{x}_6-x_6+v_5 \\ 0 &= -T_6\dot{x}_7-x_7+v_6 \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= -v_4+\begin{cases} x_1+A_5x_2+A_6\dot{x}_2 & n>0 \\ u & n=0 \end{cases} \\ 0 &= \begin{cases} -T_2(v_5-x_5)+T_1(v_4-x_5) & T_2\ne0 \\ -v_5+v_4 & T_2=0 \end{cases} \\ 0 &= \begin{cases} -T_4(v_6-x_6)+T_3(v_5-x_6) & T_4\ne0 \\ -v_6+v_5 & T_4=0 \end{cases} \\ 0 &= \begin{cases} -T_6v_7+K_\mathrm{s}T_5(v_6-x_7) & T_6\ne0 \\ -v_7+K_\mathrm{s}v_6 & T_6=0 \end{cases} \\ 0 &= -V_{\mathrm{ss}}+\text{clamp}(v_7;L_\mathrm{s}^{\min},L_\mathrm{s}^{\max}) \end{aligned}\end{split}\]

External Equations

None.

Initialization

The initial input determines the steady state; all derivatives initialize to zero.

\[\begin{split}\begin{aligned} x_1 &\leftarrow \begin{cases}u & n>0 \\ 0 & n=0\end{cases} \\ x_2,x_3,x_4 &\leftarrow 0 \\ x_5,x_6,x_7,v_4,v_5,v_6 &\leftarrow u \\ v_7 &\leftarrow \begin{cases}0 & T_6\ne0 \\ K_\mathrm{s}u & T_6=0\end{cases} \\ V_{\mathrm{ss}} &\leftarrow \text{clamp}(v_7;L_\mathrm{s}^{\min},L_\mathrm{s}^{\max}) \end{aligned}\end{split}\]

Monitors

Monitor	Units	Description	Note
vss	[p.u.]	Limited stabilizer signal	\(V_{\mathrm{ss}}\); model output