Underexcitation Limiter (UEL2C)

IEEE 421.5 Type UEL2C underexcitation limiter: keeps the generator out of the underexcited region where it would lose steady-state stability or overheat the stator end. The active power PT, normalized to rated voltage by F1 = 1/u^K1 and filtered, sets the reactive-power limit Q' from a piecewise-linear P-Q characteristic (up to six segments); F2 = u^K2 scales it back to QREF. The error QREF - QT (times a fixed gain Kfix or the automatic gain Kadj) drives a PI controller and two lead-lags. While the machine stays inside its capability the error is negative, the integrator rests at VUImin and VUEL = 0. Wire Vt, Pe and Qe from the synchronous machine and VUEL to the exciter's VUEL input. Pairs with the machine for load-flow initialization.

Category: Control / Excitation

Overview

The Type UEL2C underexcitation limiter (IEEE Std 421.5-2016, clause 11.4) keeps a generator from absorbing more reactive power than it safely can. Run too far underexcited, a machine loses steady-state stability margin and overheats the stator end region. UEL2C compares the reactive power against a limit that depends on the active power and the terminal voltage, and raises the excitation through VUEL once the machine crosses that limit.

Inside the machine's capability the limiter is idle and VUEL = 0.

Block diagram

The model has four parts.

1. Voltage normalization. The filtered terminal voltage $V_{TF}=V_T/(1+s{T}_{UV})$ passes a bias logic that gives $u$: $u=V_{TF}$ above 1 pu, $u=1$ between $V_{bias}$ and 1 pu, and $u=V_{TF}/V_{bias}$ below $V_{bias}$. Setting $V_{bias}=1$ disables the bias ($u=V_{TF}$). Then $F_1=1/u^{K_1}$ and $F_2=u^{K_2}$.

2. Limiting characteristic. The active power, normalized to rated voltage and filtered, $P^{\prime }=F_1{P}_T/(1+s{T}_{UP})$, indexes a piecewise-linear curve $Q^{\prime }(P^{\prime })$ through up to seven points $(P_0,Q_0)\dots (P_6,Q_6)$. Beyond the first and last points the end segments are extrapolated. The limit is scaled back to the actual voltage: $Q_{REF}=F_2{Q}^{\prime }/(1+s{T}_{Qref})$. With $K_1=K_2=2$ the limit follows $V^2$, like the machine's own steady-state stability limit.

3. Error and gain. The error $V_{Uerr}=Q_{REF}-Q_T/(1+s{T}_{UQ})$ is positive when the machine absorbs more than the limit allows. It is multiplied by a fixed gain $K_{fix}$, or by the automatic gain

   $$K_{adj}=\frac{V_T^2/X_q+Q_T}{\sqrt{{(V_T^2/X_q+Q_T)}^2+P_T^2}}$$

   filtered by $1/(1+s{T}_{adj})$, which reduces the loop gain at high active power. The automatic gain is kept between 0 and 1.

4. Controller. A PI controller $K_{UL}+K_{UI}/s$ limited to $[V_{UImin},V_{UImax}]$ and two limited lead-lags produce VUEL.

Limits of this implementation

The standard's optional exciter-feedback term $K_{UF}{V}_F$ and the $K_{FB}/(1+s{T}_{UL})$ feedback path are not modelled. Both apply only to particular exciter pairings and are zero in the standard's sample data.

Initialization

Inside the capability the error is negative, so the integrator rests on $V_{UImin}$ and VUEL sits at its lower limit (0 with the default data). The limiter needs the machine's terminal voltage, active and reactive power at $t=0$; the load-flow auto-initialization fills VT(0), PT(0) and QT(0) in from the machine that feeds the inputs. The first simulation step uses those seeds in place of the inputs, so the filters start on the operating point.

The machine's electrical outputs carry a short transient right after $t=0$ while its stator currents build up. With $T_{UQ}=0$ a deep dip in Qe during that transient can briefly cross the limit; add a small $T_{UQ}$ if that matters for your study.

Wiring

• Vt, Pe, Qe ← the synchronous machine's Vt, Pe and Qe outputs (pu on the machine base, generator convention: Qe < 0 is absorbing).
• VUEL → the exciter's VUEL input (enable UEL input on the exciter's Aux Inputs tab).

Enable the monitors on the Monitoring tab to record VUEL, the limit QREF and the error VUerr.

Ports

Port	Kind	Signal	Notes
Vt	Control input	Real
Pe	Control input	Real
Qe	Control input	Real
VUEL	Control output	Real

Parameters

Inputs

Parameter	Type	Default	Units	Description	ID
TUV	Number	5	s (also ms)	Terminal-voltage filter time constant 1/(1+sTUV).	TUV
Vbias	Number	1	—	Voltage-bias threshold: u = VTF above 1 pu, u = 1 between Vbias and 1, u = VTF/Vbias below Vbias. Vbias = 1 disables the bias (u = VTF).	Vbias
K1	Choice: 0 / 1 / 2	2	—	Voltage exponent of F1 = 1/u^K1 that normalizes the active power to rated voltage.	K1
K2	Choice: 0 / 1 / 2	2	—	Voltage exponent of F2 = u^K2 that scales the reactive limit back to the actual voltage.	K2
TUP	Number	5	s (also ms)	Active-power filter time constant of F1*PT/(1+sTUP).	TUP
TUQ	Number	0	s (also ms)	Reactive-power filter time constant 1/(1+sTUQ).	TUQ
TQref	Number	0	s (also ms)	Reactive-reference filter time constant of F2*Q'/(1+sTQref). 0 passes it through.	TQref

Characteristic

Parameter	Type	Default	Units	Description	ID
Points	Choice: 2 / 3 / 4 / 5 / 6 / 7	5	—	Number of points (P, Q) used to define the limiting characteristic (2 to 7, i.e. 1 to 6 segments). Beyond the first and last points the end segments are extrapolated.	n_points
P0	Number	0	—	Active power of characteristic point 0 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P0
Q0	Number	-0.31	—	Reactive power of characteristic point 0 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q0
P1	Number	0.3	—	Active power of characteristic point 1 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P1
Q1	Number	-0.31	—	Reactive power of characteristic point 1 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q1
P2	Number	0.6	—	Active power of characteristic point 2 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P2
Q2	Number	-0.28	—	Reactive power of characteristic point 2 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q2
P3	Number	0.9	—	Active power of characteristic point 3 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P3
Q3	Number	-0.21	—	Reactive power of characteristic point 3 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q3
P4	Number	1.02	—	Active power of characteristic point 4 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P4
Q4	Number	0	—	Reactive power of characteristic point 4 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q4
P5	Number	1.1	—	Active power of characteristic point 5 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P5
Q5	Number	0.2	—	Reactive power of characteristic point 5 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q5
P6	Number	1.2	—	Active power of characteristic point 6 (pu on the machine base, normalized to rated voltage). Points must increase in P.	P6
Q6	Number	0.4	—	Reactive power of characteristic point 6 (pu, negative = absorbing). The limiter acts when the generator absorbs more than this.	Q6

Gain

Parameter	Type	Default	Units	Description	ID
SW1	Choice: Fixed gain Kfix (A) / Automatic gain Kadj (B)	Fixed gain Kfix (A)	—	Error gain: a fixed reduction Kfix (position A), or the automatic dispatch-dependent gain Kadj filtered by 1/(1+sTadj) (position B).	SW1
Kfix	Number	1	—	Fixed gain applied to the error. 1 = no gain reduction.	Kfix
Tadj	Number	3	s (also ms)	Filter time constant of the automatic gain Kadj.	Tadj
Xq	Number	1.8	—	Machine quadrature-axis synchronous reactance used by Kadj = (VT^2/Xq + QT) / sqrt((VT^2/Xq + QT)^2 + PT^2) (pu).	Xq

Controller

Parameter	Type	Default	Units	Description	ID
KUL	Number	0.8	—	Proportional gain of the PI controller KUL + KUI/s.	KUL
KUI	Number	0.5	—	Integral gain of the PI controller (non-windup).	KUI
VUImax	Number	0.25	—	Upper limit of the PI output.	VUImax
VUImin	Number	0	—	Lower limit of the PI output. 0 keeps the limiter from acting outside its region.	VUImin
TU1	Number	0	s (also ms)	Lead-lag time constant TU1 of the output stages (1+sTU1)/(1+sTU2) and (1+sTU3)/(1+sTU4). Equal lead and lag (or both 0) bypasses a stage.	TU1
TU2	Number	0	s (also ms)	Lead-lag time constant TU2 of the output stages (1+sTU1)/(1+sTU2) and (1+sTU3)/(1+sTU4). Equal lead and lag (or both 0) bypasses a stage.	TU2
VUELmax2	Number	99	—	Upper limit after the first lead-lag.	VUELmax2
VUELmin2	Number	-99	—	Lower limit after the first lead-lag.	VUELmin2
TU3	Number	0	s (also ms)	Lead-lag time constant TU3 of the output stages (1+sTU1)/(1+sTU2) and (1+sTU3)/(1+sTU4). Equal lead and lag (or both 0) bypasses a stage.	TU3
TU4	Number	0	s (also ms)	Lead-lag time constant TU4 of the output stages (1+sTU1)/(1+sTU2) and (1+sTU3)/(1+sTU4). Equal lead and lag (or both 0) bypasses a stage.	TU4
VUELmax1	Number	0.25	—	Upper limit of the output VUEL.	VUELmax1
VUELmin1	Number	0	—	Lower limit of the output VUEL. 0 for a limiter that can only raise excitation.	VUELmin1

Monitoring

Parameter	Type	Default	Units	Description	ID
Monitor VUEL	Choice: Off / On	Off	—	Record the limiter output VUEL (pu) sent to the exciter's VUEL input as a named observable.	measure_vuel
Monitor QREF	Choice: Off / On	Off	—	Record the reactive-power limit QREF (pu) the characteristic gives at the present P and V as a named observable.	measure_qref
Monitor VUerr	Choice: Off / On	Off	—	Record the error VUerr = QREF - QTF (pu); the limiter acts when it turns positive as a named observable.	measure_uerr

Signal Names

Parameter	Type	Default	Units	Description	ID
VUEL name	Text	(empty)	—	Observable name for VUEL. Blank = skip.	vuel_name
QREF name	Text	(empty)	—	Observable name for QREF. Blank = skip.	qref_name
VUerr name	Text	(empty)	—	Observable name for VUerr. Blank = skip.	uerr_name

Initialization

Parameter	Type	Default	Units	Description	ID
VT(0)	Number	1	—	Initial terminal-voltage magnitude (pu). Seeds the voltage filter and stands in for Vt on the first step. Normally set by the load-flow auto-initialization.	vt_init
PT(0)	Number	0	—	Initial active power (pu on the machine base). Seeds the P filter and stands in for PT on the first step. Normally set by the load-flow auto-initialization.	p_init
QT(0)	Number	0	—	Initial reactive power (pu on the machine base). Seeds the Q filters and stands in for QT on the first step. Normally set by the load-flow auto-initialization.	q_init

Observables

Default name	Kind	Recorded when	Description	ID
set by VUEL name	Signal	Monitor VUEL is on	The limiter output VUEL (pu) sent to the exciter's VUEL input.	VUEL
set by QREF name	Signal	Monitor QREF is on	The reactive-power limit QREF (pu) the characteristic gives at the present P and V.	qref
set by VUerr name	Signal	Monitor VUerr is on	The error VUerr = QREF - QTF (pu); the limiter acts when it turns positive.	uerr