33Standard IEEE power system stabilizer: 4th-order notch filter, two lead–lag
44blocks, washout, and output limiter.
55
6+ Notes:
7+ - $V_ {cl}$, $V_ {cu}$, and $T_ {delay}$ are accepted for input-format
8+ compatibility but are not modeled.
9+ - A zero denominator time constant bypasses its corresponding lead–lag or
10+ washout block.
11+
612## Block Diagram
713
814![ ] ( ../../../../../docs/Figures/stabilizer_ieeest_diagram.png )
@@ -11,40 +17,68 @@ Figure 1: Stabilizer IEEEST model. Figure courtesy of [PowerWorld](https://www.p
1117
1218## Model Parameters
1319
14- Symbol | Units | Description | Typical Value
15- ------------|--------|--------------------------------------|--------------
16- $A_1$ | [ s] | Notch denominator coefficient | 1.013
17- $A_2$ | [ s²] | Notch denominator coefficient | 0.013
18- $A_3$ | [ s] | Notch denominator coefficient | 0.0
19- $A_4$ | [ s²] | Notch denominator coefficient | 0.0
20- $A_5$ | [ s] | Notch numerator coefficient | 1.013
21- $A_6$ | [ s²] | Notch numerator coefficient | 0.113
22- $T_1$ | [ s] | Lead–lag 1 numerator time constant | 0.0
23- $T_2$ | [ s] | Lead–lag 1 denominator time constant | 0.02
24- $T_3$ | [ s] | Lead–lag 2 numerator time constant | 0.0
25- $T_4$ | [ s] | Lead–lag 2 denominator time constant | 0.0
26- $T_5$ | [ s] | Washout numerator time constant | 1.65
27- $T_6$ | [ s] | Washout denominator time constant | 1.65
28- $K_s$ | [ p.u.] | Stabilizer gain | 3.0
29- $L_s^{\min}$ | [ p.u.] | Minimum stabilizer output limit | -0.1
30- $L_s^{\max}$ | [ p.u.] | Maximum stabilizer output limit | 0.1
31-
32- The IEEE 421.5 IEEEST also defines a cutout window ($V_ {cl}$, $V_ {cu}$) and an
33- input delay ($T_ {delay}$). These parameters are accepted for input-format
34- compatibility but are not modeled here.
35-
36- ### Derived Parameters
20+ Symbol | Units | JSON | Description | Typical Value | Note
21+ -------------|--------|----------|--------------------------------------|---------------|------
22+ $A_1$ | [ s] | ` A1 ` | Notch denominator coefficient | 1.013 |
23+ $A_2$ | [ s²] | ` A2 ` | Notch denominator coefficient | 0.013 |
24+ $A_3$ | [ s] | ` A3 ` | Notch denominator coefficient | 0.0 |
25+ $A_4$ | [ s²] | ` A4 ` | Notch denominator coefficient | 0.0 |
26+ $A_5$ | [ s] | ` A5 ` | Notch numerator coefficient | 1.013 |
27+ $A_6$ | [ s²] | ` A6 ` | Notch numerator coefficient | 0.113 |
28+ $T_1$ | [ s] | ` T1 ` | Lead–lag 1 numerator time constant | 0.0 |
29+ $T_2$ | [ s] | ` T2 ` | Lead–lag 1 denominator time constant | 0.02 |
30+ $T_3$ | [ s] | ` T3 ` | Lead–lag 2 numerator time constant | 0.0 |
31+ $T_4$ | [ s] | ` T4 ` | Lead–lag 2 denominator time constant | 0.0 |
32+ $T_5$ | [ s] | ` T5 ` | Washout numerator time constant | 1.65 |
33+ $T_6$ | [ s] | ` T6 ` | Washout denominator time constant | 1.65 |
34+ $K_s$ | [ p.u.] | ` Ks ` | Stabilizer gain | 3.0 |
35+ $L_s^{\min}$ | [ p.u.] | ` Lsmin ` | Minimum stabilizer output limit | -0.1 |
36+ $L_s^{\max}$ | [ p.u.] | ` Lsmax ` | Maximum stabilizer output limit | 0.1 |
37+ $V_ {cl}$ | [ p.u.] | ` Vcl ` | Lower input cutout threshold | 0.0 | Accepted but not modeled
38+ $V_ {cu}$ | [ p.u.] | ` Vcu ` | Upper input cutout threshold | 0.0 | Accepted but not modeled
39+ $T_ \text{delay}$ | [ s] | ` Tdelay ` | Input delay | 0.0 | Accepted but not modeled
40+
41+ ### Parameter Validation
42+
43+ The fixed realization rejects a first-order-only notch denominator:
44+
45+ ``` math
46+ \begin{aligned}
47+ a_2 \ne 0 \lor a_3 \ne 0 \lor a_4 \ne 0 \lor a_1 = 0
48+ \end{aligned}
49+ ```
50+
51+ ### Model Derived Parameters
52+
53+ The notch-filter denominator expands to:
3754
3855``` math
3956\begin{aligned}
40- a_0 &= 1 \\
4157a_1 &= A_1 + A_3 \\
4258a_2 &= A_2 + A_4 + A_1 A_3 \\
4359a_3 &= A_1 A_4 + A_2 A_3 \\
4460a_4 &= A_2 A_4
4561\end{aligned}
4662```
4763
64+ The binary DAE selectors choose the active notch-filter order:
65+
66+ ``` math
67+ \begin{aligned}
68+ \delta_1 &= \delta_2 =
69+ \begin{cases}
70+ 1 & a_2 \ne 0 \lor a_3 \ne 0 \lor a_4 \ne 0 \\
71+ 0 & \text{otherwise}
72+ \end{cases}
73+ \\
74+ \delta_3 &=
75+ \begin{cases}
76+ 1 & a_3 \ne 0 \lor a_4 \ne 0 \\
77+ 0 & \text{otherwise}
78+ \end{cases}
79+ \end{aligned}
80+ ```
81+
4882## Model Variables
4983
5084### Internal Variables
@@ -58,6 +92,9 @@ $x_5$ | [-] | Lead–lag 1 state
5892$x_6$ | [ -] | Lead–lag 2 state
5993$x_7$ | [ -] | Washout state
6094
95+ For reduced-order notch filters, unused notch states remain in the fixed
96+ component state vector and are pinned by algebraic residuals.
97+
6198#### Algebraic
6299
63100Symbol | Units | Description
@@ -82,10 +119,10 @@ $u$ | [p.u.] | Stabilizer input signal
82119
83120``` math
84121\begin{aligned}
85- 0 &= -\dot{x}_1 + x_2 \\
86- 0 &= -\dot{x}_2 + x_3 \\
87- 0 &= -\dot{x}_3 + x_4 \\
88- 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 \\
122+ 0 &= -\delta_1\ dot{x}_1 + x_2 \\
123+ 0 &= -\delta_2\ dot{x}_2 + x_3 \\
124+ 0 &= -\delta_3\ dot{x}_3 + x_4 \\
125+ 0 &= -a_4 \dot{x}_4 - x_1 - a_1x_2 - a_2x_3 - a_3x_4 + u \\
891260 &= -T_2 \dot{x}_5 - x_5 + v_4 \\
901270 &= -T_4 \dot{x}_6 - x_6 + v_5 \\
911280 &= -T_6 \dot{x}_7 - x_7 + v_6
@@ -96,10 +133,22 @@ $u$ | [p.u.] | Stabilizer input signal
96133
97134``` math
98135\begin{aligned}
99- 0 &= -v_4 + x_1 + A_5 x_2 + A_6 x_3 \\
100- 0 &= -T_2(v_5 - x_5) + T_1(v_4 - x_5) \\
101- 0 &= -T_4(v_6 - x_6) + T_3(v_5 - x_6) \\
102- 0 &= -T_6 v_7 + K_s T_5(v_6 - x_7) \\
136+ 0 &= -v_4 + x_1 + A_5x_2 + A_6x_3 \\
137+ 0 &=
138+ \begin{cases}
139+ -T_2(v_5 - x_5) + T_1(v_4 - x_5) & T_2 \ne 0 \\
140+ v_4 - v_5 & T_2 = 0
141+ \end{cases} \\
142+ 0 &=
143+ \begin{cases}
144+ -T_4(v_6 - x_6) + T_3(v_5 - x_6) & T_4 \ne 0 \\
145+ v_5 - v_6 & T_4 = 0
146+ \end{cases} \\
147+ 0 &=
148+ \begin{cases}
149+ -T_6 v_7 + K_s T_5(v_6 - x_7) & T_6 \ne 0 \\
150+ K_s v_6 - v_7 & T_6 = 0
151+ \end{cases} \\
1031520 &= -V_{ss} + \text{clamp}(v_7, L_s^{\min}, L_s^{\max})
104153\end{aligned}
105154```
@@ -109,6 +158,27 @@ The output limiter uses GridKit's smooth
109158
110159## Initialization
111160
112- All states and their derivatives initialize to zero. The stabilizer comes
113- online at rest and produces signal only in response to deviations in the input
114- $u$.
161+ States and derivatives initialize to the steady state implied by the attached
162+ input $u$:
163+
164+ ``` math
165+ \begin{aligned}
166+ x_1 &= v_4 = x_5 = v_5 = x_6 = v_6 = x_7 = u \\
167+ x_2 &= x_3 = x_4 = 0 \\
168+ v_7 &=
169+ \begin{cases}
170+ K_su & T_6 = 0 \\
171+ 0 & \text{otherwise}
172+ \end{cases}
173+ &
174+ V_{ss} &= \text{clamp}(v_7, L_s^{\min}, L_s^{\max})
175+ \end{aligned}
176+ ```
177+
178+ All internal derivatives initialize to zero.
179+
180+ ## Model Outputs
181+
182+ Output | Units | Description | Note
183+ -------|--------|---------------------------|-----
184+ ` vss ` | [ p.u.] | Limited stabilizer signal | Exported through ` output ` when assigned
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