InverseTimeRelay

An inverse-time overcurrent relay with optional lockout.

Notes

• Designed to measure a BranchBreakers terminal current and to drive a BranchBreakers trip1 or trip2 input.

• Above pickup, the trip rises once \(\int t_\mathrm{op}^{-1}\,dt = 1\), the IEEE C37.112[1] operate criterion.

• With lockout, the relay commits to trip once \(x\) passes \(1/2\), so it may measure the current it interrupts.

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(I_\mathrm{pickup}\)	[p.u.]	Ipickup	Pickup current		Required
\(D\)	[-]	TD	Time dial		Required
\(A\)	[s]	A	Curve coefficient		Required
\(B\)	[s]	B	Curve offset		Required
\(\alpha\)	[-]	alpha	Curve exponent		Required
\(t_r\)	[s]	tr	Reset constant	0.0	0 for instantaneous reset
\(s_\mathrm{lock}\)	[boolean]	Lockout	Lockout selector	true	true = lockout, false = self-reset

IEEE C37.112 curves:

Curve	A	B	alpha	tr
Moderately inverse	0.0515	0.1140	0.02	4.85
Very inverse	19.61	0.491	2.0	21.6
Extremely inverse	28.2	0.1217	2.0	29.1

Parameter Validation

\[\begin{split}\begin{aligned} I_\mathrm{pickup} &> 0 \\ D &> 0 \\ A &> 0 \\ B &\ge 0 \\ \alpha &> 0 \\ t_r &\ge 0 \end{aligned}\end{split}\]

All real parameters must be finite.

Model Derived Parameters

\[\begin{split}\begin{aligned} \epsilon_T &= 10^{-3}\ \mathrm{s} \\ T_\mathrm{reset} &= \max(D t_r, \epsilon_T) \end{aligned}\end{split}\]

Model Ports

Name	Port	Init	Description
ir	Input	Known	Required measured current, real component
ii	Input	Known	Required measured current, imaginary component
trip	Output	Known	Required trip command

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(x\)	[-]	Relay latch

Algebraic

Symbol	Units	Description	Note
\(s\)	[-]	Trip command	trip

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(I_r\)	[p.u.]	Measured current, real component	ir
\(I_i\)	[p.u.]	Measured current, imaginary component	ii

Model Equations

Smooth functions: above, latch, logistic (\(\sigma\)), max, min.

For readability, define:

\[\begin{split}\begin{aligned} M^2 &= \dfrac{I_r^2 + I_i^2}{I_\mathrm{pickup}^2} \\ p &= \text{logistic}(M^2 - 1) \\ K &= \text{max}(M^2, 1)^{\alpha/2} - 1 \\ \dfrac{1}{t_\mathrm{op}} &= \dfrac{K}{D(A + B K)} \\ \dfrac{1}{t_\mathrm{rs}} &= \dfrac{1 - \text{min}(M^2, 1)}{T_\mathrm{reset}} \\ \dfrac{1}{T_\mathrm{latch}} &= \ln 4\left(\dfrac{p}{t_\mathrm{op}} + \dfrac{1 - p}{t_\mathrm{rs}}\right) \end{aligned}\end{split}\]

Internal Equations

Differential

\[\begin{aligned} 0 &= -\dot{x} + \dfrac{1}{T_\mathrm{latch}}\text{latch}(x, p, (1 - s_\mathrm{lock})(1 - p)) \end{aligned}\]

Algebraic

\[\begin{aligned} 0 &= -s + \text{above}(x; 3/4) \end{aligned}\]

External Equations

None.

Initialization

\[\begin{split}\begin{aligned} x &\leftarrow 0 \\ s &\leftarrow \text{above}(x; 3/4) \end{aligned}\end{split}\]

All internal derivatives initialize to zero.

Monitors

Monitor	Units	Description	Note
im	[p.u.]	Measured current magnitude	\(\sqrt{I_r^2 + I_i^2}\)
x	[-]	Relay latch	\(x\)
trip	[-]	Trip command	\(s\)

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[1]

IEEE Std C37.112-2018, IEEE Standard for Inverse-Time Characteristics Equations for Overcurrent Relays.