REECB

REECB is a WECC renewable electrical-control model with power-factor, reactive-power, voltage, and active-power command paths for an inverter-coupled resource.[1]

Warning

GridKit does not inherit mva from the associated REGCA model. Set mva to the REGCA component base.[2]

Warning

GridKit does not yet apply the associated generator’s Governor Response Limits modes Down Only and Fixed to REECB.

Notes

• In direct-voltage mode (\(s_Q=1\), \(s_V=0\)) qext carries a terminal-voltage reference instead of a system-base reactive power.

Block Diagram

Figure 1: REECB electrical-control model. Figure courtesy of the PowerWorld REEC_B model reference.

Model Parameters

Symbol	Units	JSON	Description	Typical Value	Note
\(S^\mathrm{base}\)	[MVA]	mva	REECB component power base	Required	Must be positive
\(s_\mathrm{pf}\)	[boolean]	PfFlag	Power-factor control selector	false	true = power-factor control, false = reactive-power control
\(s_V\)	[boolean]	VFlag	Voltage-reference selector under \(s_Q=1\)	false	true = cascaded Q-PI voltage command, false = direct external voltage reference
\(s_Q\)	[boolean]	QFlag	Reactive-path selector	false	true = Volt/VAr PI control, false = reactive-current lag
\(s_\mathrm{pq}\)	[boolean]	Pqflag	Converter current-priority selector	false	true = P priority, false = Q priority
\(T_\mathrm{rv}\)	[s]	Trv	Voltage-measurement filter time constant	0.02	State 1 in Fig. 1
\(T_\mathrm{p}\)	[s]	Tp	Electrical-power measurement filter time constant	0.0	State 2 in Fig. 1
\(V^\mathrm{ref}\)	[p.u.]	Vref0	Reactive-current-injection voltage reference	\(V_T\)	Initialized from terminal voltage when omitted
\(V_\mathrm{dip}\)	[p.u.]	Vdip	Low-voltage threshold for the voltage-band gate	0.85
\(V_\mathrm{up}\)	[p.u.]	Vup	High-voltage threshold for the voltage-band gate	1.15
\(D_1^\mathrm{db}\)	[p.u.]	dbd1	Lower deadband threshold for voltage-error response	0.0
\(D_2^\mathrm{db}\)	[p.u.]	dbd2	Upper deadband threshold for voltage-error response	0.0
\(K_\mathrm{qv}\)	[p.u.]	kqv	Reactive-current injection gain	5.0
\(I_{q,\mathrm{inj}}^{\min}\)	[p.u.]	Iql1	Minimum reactive-current injection	-1.1
\(I_{q,\mathrm{inj}}^{\max}\)	[p.u.]	Iqh1	Maximum reactive-current injection	1.1
\(Q^{\max}\)	[p.u.]	Qmax	Maximum reactive-power control output	0.436
\(Q^{\min}\)	[p.u.]	Qmin	Minimum reactive-power control output	-0.436
\(K_\mathrm{qp}\)	[p.u.]	Kqp	Reactive-power controller proportional gain	0.0
\(K_\mathrm{qi}\)	[p.u./s]	Kqi	Reactive-power controller integral gain	0.1
\(V^{\max}\)	[p.u.]	Vmax	Maximum voltage-control output	1.1
\(V^{\min}\)	[p.u.]	Vmin	Minimum voltage-control output	0.9
\(K_\mathrm{vp}\)	[p.u.]	Kvp	Voltage controller proportional gain	18.0
\(K_\mathrm{vi}\)	[p.u./s]	Kvi	Voltage controller integral gain	5.0
\(T_\mathrm{iq}\)	[s]	Tiq	Reactive-current command lag time constant	0.02	State 5 in Fig. 1
\(T_\mathrm{pord}\)	[s]	Tpord	Active-power order filter time constant	0.02	State 6 in Fig. 1
\(R_P^{\max}\)	[p.u./s]	dPmax	Positive active-power order ramp-rate limit	99.0
\(R_P^{\min}\)	[p.u./s]	dPmin	Negative active-power order ramp-rate limit	-99.0
\(P^{\max}\)	[p.u.]	Pmax	Maximum active-power order	1.0
\(P^{\min}\)	[p.u.]	Pmin	Minimum active-power order	0.0
\(I^{\max}\)	[p.u.]	Imax	Maximum converter current	1.3

Real-valued parameters accept real or integer JSON values; selectors require Boolean JSON values.

Parameter Validation

A valid REECB parameter set must satisfy the following conditions:

\[\begin{split}\begin{aligned} S^\mathrm{base} &> 0 \\ T_\mathrm{rv},T_\mathrm{p},T_\mathrm{iq},T_\mathrm{pord} &\ge 0 \\ V_\mathrm{dip} &< V_\mathrm{up} \\ D_1^\mathrm{db} &\le 0 \le D_2^\mathrm{db} \\ K_\mathrm{qv},K_\mathrm{qp},K_\mathrm{qi},K_\mathrm{vp},K_\mathrm{vi} &\ge 0 \\ I_{q,\mathrm{inj}}^{\min} &\le I_{q,\mathrm{inj}}^{\max} \\ Q^{\min} &\le Q^{\max} \\ V^{\min} &\le V^{\max} \\ R_P^{\min} &< 0 < R_P^{\max} \\ P^{\min} &\le P^{\max} \\ I^{\max} &> 0 \end{aligned}\end{split}\]

Enabling both PfFlag and QFlag logs an atypical-configuration warning.

Model Derived Parameters

Let \(\epsilon_T=10^{-3}\ \mathrm{s}\). A time constant below \(\epsilon_T\) is raised to that floor in place, so every equation below uses the raised value:

\[\begin{split}\begin{aligned} T_x &\leftarrow \max(T_x,\epsilon_T), && x\in\{\mathrm{rv},\mathrm{p},\mathrm{iq},\mathrm{pord}\} \\ s_\mathrm{pf}^\mathrm{off} &= 1 - s_\mathrm{pf} \\ s_Q^\mathrm{off} &= 1 - s_Q \\ s_Q^\mathrm{PI} &= s_Q s_V \\ s_V^\mathrm{ref} &= s_Q(1-s_V) \\ s_Q^\mathrm{ref} &= 1 - s_V^\mathrm{ref} \\ s_\mathrm{pq}^\mathrm{off} &= 1 - s_\mathrm{pq} \\ k_\mathrm{base} &= \dfrac{S^\mathrm{sys}}{S^\mathrm{base}} \end{aligned}\end{split}\]

Multiplying by \(k_\mathrm{base}\) converts system base to component base.

Model Ports

Name	Port	Init	Description
bus	Bus	Known	Terminal-bus voltage
pe	Input	Known	Active-power feedback
qgen	Input	Known	Reactive-power feedback
qext	Input	Unknown	Volt/VAr reference
pfaref	Input	Unknown	Power-factor angle reference
pref	Input	Unknown	Active-power reference
iqcmd	Output	Known	Reactive-current command
ipcmd	Output	Known	Active-current command

bus is required; signal ports are optional and must be linked when attached. Known values are seeded before initialization and preserved. Unknown inputs are initialized in attached signal storage or held constant when unattached.

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(V^\mathrm{meas}\)	[p.u.]	Filtered terminal voltage	State 1 in Fig. 1; Source label: Vmeas
\(P^\mathrm{meas}\)	[p.u.]	Filtered electrical power	State 2 in Fig. 1; Source label: Pmeas; component base
\(x_Q^\mathrm{PI}\)	[p.u.]	Reactive-power PI controller state	State 3 in Fig. 1; Source label: PIQ
\(x_V^\mathrm{PI}\)	[p.u.]	Voltage-control PI controller state	State 4 in Fig. 1; Source label: PIV; component-base current
\(Q_V\)	[p.u.]	Reactive-current command lag state	State 5 in Fig. 1; Source label: Q_V; component base
\(P^\mathrm{ord}\)	[p.u.]	Filtered active-power order	State 6 in Fig. 1; Source label: Pord; component base

Algebraic

Symbol	Units	Description	Note
\(V_T\)	[p.u.]	Terminal voltage magnitude
\(V_\mathrm{safe}^\mathrm{meas}\)	[p.u.]	Safe measured voltage
\(s_\mathrm{dip}\)	[-]	Voltage-band gate
\(I_q^\mathrm{inj}\)	[p.u.]	Reactive-current injection	Component base
\(Q^\mathrm{ref}\)	[p.u.]	Reactive-power reference	Component base
\(e_Q\)	[p.u.]	Reactive-power error	Component base
\(V_Q^\mathrm{PI}\)	[p.u.]	Reactive-power PI output
\(e_V^\mathrm{PI}\)	[p.u.]	Voltage-control error
\(r_P^\mathrm{ord}\)	[p.u./s]	Limited active-power order rate	Component base
\(I_L^\mathrm{cap}\)	[p.u.]	Off-axis current capacity	Component base
\(I_q^{\max}\)	[p.u.]	Reactive-current limit	Component base
\(I_p^{\max}\)	[p.u.]	Active-current limit	Component base
\(I_q^\mathrm{base}\)	[p.u.]	Voltage-controller current	Component base
\(I_q^\mathrm{raw}\)	[p.u.]	Pre-limit reactive-current command	Component base
\(I_q^\mathrm{cmd}\)	[p.u.]	Reactive-current command output	System base
\(I_p^\mathrm{cmd}\)	[p.u.]	Active-current command output	System base

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(V_r\)	[p.u.]	Terminal voltage, real component	Bus input
\(V_i\)	[p.u.]	Terminal voltage, imaginary component	Bus input
\(P_\mathrm{e}\)	[p.u.]	Electrical active-power feedback	Optional signal port pe; system base
\(Q^\mathrm{gen}\)	[p.u.]	Reactive-power feedback	Optional signal port qgen; system base
\(Q^\mathrm{ext}\)	[p.u.]	External Volt/VAr reference	Optional signal port qext; terminal voltage in direct-voltage mode
\(\phi^\mathrm{ref}\)	[rad]	Power-factor angle reference	Optional signal port pfaref
\(P^\mathrm{ref}\)	[p.u.]	External active-power reference	Optional signal port pref; system base

Model Equations

Smooth functions: antiwindup, clamp, deadband2, inside, max.

Internal Equations

Differential

\[\begin{split}\begin{aligned} 0 &= -\dot{V}^\mathrm{meas} + \dfrac{1}{T_\mathrm{rv}}(V_T-V^\mathrm{meas}) \\ 0 &= -\dot{P}^\mathrm{meas} + \dfrac{1}{T_\mathrm{p}}(k_\mathrm{base}P_\mathrm{e}-P^\mathrm{meas}) \\ 0 &= -\dot{x}_Q^\mathrm{PI} + s_Q^\mathrm{PI}s_\mathrm{dip}\,\text{antiwindup}(K_\mathrm{qp}e_Q+x_Q^\mathrm{PI},K_\mathrm{qi}e_Q;\,V^{\min},V^{\max}) \\ 0 &= -\dot{x}_V^\mathrm{PI} + s_Qs_\mathrm{dip}\,\text{antiwindup}(K_\mathrm{vp}e_V^\mathrm{PI}+x_V^\mathrm{PI},K_\mathrm{vi}e_V^\mathrm{PI};\,-I_q^{\max},I_q^{\max}) \\ 0 &= -\dot{Q}_V + \dfrac{1}{T_\mathrm{iq}}s_Q^\mathrm{off}s_\mathrm{dip}\left(\dfrac{Q^\mathrm{ref}}{V_\mathrm{safe}^\mathrm{meas}}-Q_V\right) \\ 0 &= -\dot{P}^\mathrm{ord} + s_\mathrm{dip}\,\text{antiwindup}(P^\mathrm{ord},r_P^\mathrm{ord};\,P^{\min},P^{\max}) \end{aligned}\end{split}\]

Algebraic

\[\begin{split}\begin{aligned} 0 &= -V_T^2+V_r^2+V_i^2 \\ 0 &= -V_\mathrm{safe}^\mathrm{meas}+\text{max}(V^\mathrm{meas},0.01) \\ 0 &= -s_\mathrm{dip}+\text{inside}(V_T;\,V_\mathrm{dip},V_\mathrm{up}) \\ 0 &= -I_q^\mathrm{inj}+\text{clamp}\!(K_\mathrm{qv}\text{deadband2}(V^\mathrm{ref}-V^\mathrm{meas};\,D_1^\mathrm{db},D_2^\mathrm{db});\,I_{q,\mathrm{inj}}^{\min},I_{q,\mathrm{inj}}^{\max}) \\ 0 &= -Q^\mathrm{ref}+s_Q^\mathrm{ref}(s_\mathrm{pf}P^\mathrm{meas}\tan(\phi^\mathrm{ref})+s_\mathrm{pf}^\mathrm{off}k_\mathrm{base}Q^\mathrm{ext}) \\ 0 &= -e_Q+\text{clamp}(Q^\mathrm{ref};\,Q^{\min},Q^{\max})-k_\mathrm{base}Q^\mathrm{gen} \\ 0 &= -V_Q^\mathrm{PI}+\text{clamp}(K_\mathrm{qp}e_Q+x_Q^\mathrm{PI};\,V^{\min},V^{\max}) \\ 0 &= -e_V^\mathrm{PI}+s_Q^\mathrm{PI}V_Q^\mathrm{PI}+s_V^\mathrm{ref}Q^\mathrm{ext}-s_QV^\mathrm{meas} \\ 0 &= -r_P^\mathrm{ord}+\text{aslew}\!\left(\dfrac{k_\mathrm{base}P^\mathrm{ref}-P^\mathrm{ord}}{T_\mathrm{pord}};\,R_P^{\min},R_P^{\max}\right) \\ 0 &= -I_L^\mathrm{cap}+\text{sqrtramp}\big((I^{\max}-I^\mathrm{high})(I^{\max}+I^\mathrm{high})\big) \\ 0 &= -I_q^{\max}+s_\mathrm{pq}I_L^\mathrm{cap}+s_\mathrm{pq}^\mathrm{off}I^{\max} \\ 0 &= -I_p^{\max}+s_\mathrm{pq}I^{\max}+s_\mathrm{pq}^\mathrm{off}I_L^\mathrm{cap} \\ 0 &= -I_q^\mathrm{base}+\text{clamp}(K_\mathrm{vp}e_V^\mathrm{PI}+x_V^\mathrm{PI};\,-I_q^{\max},I_q^{\max}) \\ 0 &= -I_q^\mathrm{raw}+s_QI_q^\mathrm{base}+s_Q^\mathrm{off}Q_V+I_q^\mathrm{inj} \\ 0 &= -k_\mathrm{base}I_q^\mathrm{cmd}+\text{clamp}(I_q^\mathrm{raw};\,-I_q^{\max},I_q^{\max}) \\ 0 &= -k_\mathrm{base}I_p^\mathrm{cmd}+\text{clamp}\left(\dfrac{P^\mathrm{ord}}{V_\mathrm{safe}^\mathrm{meas}};\,0,I_p^{\max}\right) \end{aligned}\end{split}\]

Here \(I^\mathrm{high}=s_\mathrm{pq}k_\mathrm{base}I_p^\mathrm{cmd} +s_\mathrm{pq}^\mathrm{off}k_\mathrm{base}I_q^\mathrm{cmd}\) and \(\epsilon_\mathrm{init}=100\epsilon_\mathrm{machine}\). Appendix B defines sqrtramp.

Appendix A defines aslew.

External Equations

None.

Initialization

REECB reconstructs a steady operating point. Arbitrary-state restart is unsupported.

Input Initialization

\[\begin{split}\begin{aligned} V_r,V_i &\leftarrow \text{terminal-bus voltage} \\ I_q^\mathrm{cmd},I_p^\mathrm{cmd} &\leftarrow \text{owned current-command variables} \\ P_\mathrm{e} &\leftarrow \text{attached active-power feedback},\quad \text{if attached} \\ Q^\mathrm{gen} &\leftarrow \text{attached reactive-power feedback},\quad \text{if attached} \end{aligned}\end{split}\]

Internal Initialization

Initialization resolves the steady-state quantities in dependency order; all internal derivatives start at zero. Initialize the component-base commands: \(I_p\leftarrow k_\mathrm{base}I_p^\mathrm{cmd}\) and \(I_q\leftarrow k_\mathrm{base}I_q^\mathrm{cmd}\). REECB’s iclamp helper inverts the smooth CommonMath clamp during initialization. At a bound, REECB uses an outward offset \(\delta=-\ln(\text{expm1}(\mu\epsilon_\mathrm{init}/2))/\mu\) to keep the clamp error within \(\epsilon_\mathrm{init}\); collapsed bounds return the bound.

\[\begin{split}\begin{aligned} V_T &\leftarrow \sqrt{V_r^2+V_i^2} \\ V^\mathrm{ref} &\leftarrow V_T,\quad \text{if omitted} \\ V^\mathrm{meas} &\leftarrow V_T \\ V_\mathrm{safe}^\mathrm{meas} &\leftarrow \text{max}(V^\mathrm{meas},0.01) \\ s_\mathrm{dip} &\leftarrow \text{inside}(V_T;\,V_\mathrm{dip},V_\mathrm{up}) \\ P_\mathrm{e} &\leftarrow V_\mathrm{safe}^\mathrm{meas}I_p^\mathrm{cmd},\quad \text{if unattached} \\ Q^\mathrm{gen} &\leftarrow V_\mathrm{safe}^\mathrm{meas}I_q^\mathrm{cmd},\quad \text{if unattached} \\ P^\mathrm{meas} &\leftarrow k_\mathrm{base}P_\mathrm{e} \\ e_V^\mathrm{db} &\leftarrow \text{deadband2}(V^\mathrm{ref}-V^\mathrm{meas};\,D_1^\mathrm{db},D_2^\mathrm{db}) \\ I_q^\mathrm{inj} &\leftarrow \text{clamp}(K_\mathrm{qv}e_V^\mathrm{db};\,I_{q,\mathrm{inj}}^{\min},I_{q,\mathrm{inj}}^{\max}) \end{aligned}\end{split}\]

Let \(I_q^\mathrm{need}\) be the smooth-clamp input magnitude required to reproduce \(I_q\), including the Volt/VAr recovery margin. Initialization raises \(I^{\max}\), if needed, to a finite limit satisfying

\[\begin{split}\begin{aligned} h &\leftarrow s_\mathrm{pq}I_p+s_\mathrm{pq}^\mathrm{off}|I_q| \\ \ell &\leftarrow s_\mathrm{pq}I_q^\mathrm{need}+s_\mathrm{pq}^\mathrm{off}I_p \\ I &\ge \max(I^{\max},h,\ell,I_q^\mathrm{need}) \\ \text{sqrtramp}\big((I-h)(I+h)\big) &\ge \ell \end{aligned}\end{split}\]

The solution initializes \(I^{\max}\leftarrow I\) and \(I_L^\mathrm{cap}\leftarrow\text{sqrtramp}\big((I-h)(I+h)\big)\). The operating point is rejected if no finite solution exists or the reactive-current injection is incompatible. Q, V, and P limits are expanded as needed; each adjustment logs a warning. This matches PowerWorld’s default Modify Limits and Run treatment of initial limit violations. This preserves the supplied operating point for steady-state initialization; subsequent simulation uses the adjusted limits.

\[\begin{split}\begin{aligned} I_q^\mathrm{raw} &\leftarrow \text{iclamp}(I_q;\,-I_q^{\max},I_q^{\max}) \\ I_q^\mathrm{ctrl} &\leftarrow I_q^\mathrm{raw}-I_q^\mathrm{inj} \\ P^\mathrm{ord} &\leftarrow V_\mathrm{safe}^\mathrm{meas}\text{iclamp}(I_p;\,0,I_p^{\max}) \\ Q^\mathrm{target} &\leftarrow \begin{cases} V_\mathrm{safe}^\mathrm{meas}I_q^\mathrm{ctrl} & s_Q=0 \\ \text{iclamp}(k_\mathrm{base}Q^\mathrm{gen};\,Q^{\min},Q^{\max}) & s_Qs_V=1 \\ 0 & \text{otherwise} \end{cases} \end{aligned}\end{split}\]

The references resolved under Output Initialization give

\[\begin{split}\begin{aligned} Q^\mathrm{ref} &\leftarrow s_Q^\mathrm{ref}(s_\mathrm{pf}P^\mathrm{meas}\tan(\phi^\mathrm{ref})+s_\mathrm{pf}^\mathrm{off}k_\mathrm{base}Q^\mathrm{ext}) \\ e_Q &\leftarrow \text{clamp}(Q^\mathrm{ref};\,Q^{\min},Q^{\max})-k_\mathrm{base}Q^\mathrm{gen} \\ x_Q^\mathrm{PI} &\leftarrow \begin{cases} \text{iclamp}(V^\mathrm{meas};\,V^{\min},V^{\max})-K_\mathrm{qp}e_Q & s_Qs_V=1 \\ 0 & s_Qs_V=0 \end{cases} \\ V_Q^\mathrm{PI} &\leftarrow \text{clamp}(K_\mathrm{qp}e_Q+x_Q^\mathrm{PI};\,V^{\min},V^{\max}) \\ e_V^\mathrm{PI} &\leftarrow s_Q^\mathrm{PI}V_Q^\mathrm{PI}+s_V^\mathrm{ref}Q^\mathrm{ext}-s_QV^\mathrm{meas} \\ x_V^\mathrm{PI} &\leftarrow \begin{cases} -K_\mathrm{vp}e_V^\mathrm{PI} & s_Q=1\ \land\ I_q^{\max}\le\epsilon_\mathrm{init} \\ \text{iclamp}(I_q^\mathrm{ctrl};\,-I_q^{\max},I_q^{\max})-K_\mathrm{vp}e_V^\mathrm{PI} & s_Q=1\ \land\ I_q^{\max}>\epsilon_\mathrm{init} \\ 0 & s_Q=0 \end{cases} \\ Q_V &\leftarrow \begin{cases} 0 & s_Q=1 \\ Q^\mathrm{ref}/V_\mathrm{safe}^\mathrm{meas} & s_Q=0 \end{cases} \\ I_q^\mathrm{base} &\leftarrow \text{clamp}(K_\mathrm{vp}e_V^\mathrm{PI}+x_V^\mathrm{PI};\,-I_q^{\max},I_q^{\max}) \\ I_q^\mathrm{raw} &\leftarrow s_QI_q^\mathrm{base}+s_Q^\mathrm{off}Q_V+I_q^\mathrm{inj} \\ r_P^\mathrm{ord} &\leftarrow 0 \end{aligned}\end{split}\]

Invalid or non-equilibrium operating points are rejected before any state, limit, latch, or signal is changed.

Output Initialization

\[\begin{split}\begin{aligned} \phi^\mathrm{ref} &\leftarrow \begin{cases} \arctan(Q^\mathrm{target}/P^\mathrm{meas}) & s_\mathrm{pf}=1\ \land\ P^\mathrm{meas}\ne0 \\ 0 & \text{otherwise} \end{cases} \\ Q^\mathrm{ext} &\leftarrow \begin{cases} V^\mathrm{meas} & s_V^\mathrm{ref}=1 \\ 0 & s_V^\mathrm{ref}=0\ \land\ s_\mathrm{pf}=1 \\ Q^\mathrm{target}/k_\mathrm{base} & s_V^\mathrm{ref}=0\ \land\ s_\mathrm{pf}=0 \end{cases} \\ P^\mathrm{ref} &\leftarrow \dfrac{P^\mathrm{ord}}{k_\mathrm{base}} \end{aligned}\end{split}\]

REECB writes the resolved references to attached signal inputs; unattached ports retain them as constant inputs.

Monitors

Monitor	Units	Description	Note
iqcmd	[p.u.]	Reactive-current command output	\(I_q^\mathrm{cmd}\) (system base)
ipcmd	[p.u.]	Active-current command output	\(I_p^\mathrm{cmd}\) (system base)
vmeas	[p.u.]	Filtered terminal voltage	\(V^\mathrm{meas}\)
pmeas	[p.u.]	Filtered electrical power	\(P^\mathrm{meas}\) (component base)

Appendix A: aslew

For \(\ell<0<u\), REECB uses

\[\text{aslew}(f;\ell,u) =\dfrac{f}{1+\rho(f/u-1)+\rho(f/\ell-1)}\]

where \(\rho\) is GridKit’s smooth ramp. With exact one-sided ramps this reduces to \(\text{clamp}(f;\ell,u)\); the smooth form preserves \(\text{aslew}(0;\ell,u)=0\).

Appendix B: sqrtramp

The off-axis capacity approaches \(\sqrt{\max(0,s)}\), where \(s=(I^{\max}-I^\mathrm{high})(I^{\max}+I^\mathrm{high})\). Let \(\kappa=\epsilon_\mathrm{init}/\mu\), with \(\mu\) GridKit’s smoothing scale. REECB uses

\[\begin{split}\begin{aligned} h_\kappa(s) &= \tfrac{1}{2}\left(s+\sqrt{s^2+\kappa^2}\right)\\ \text{sqrtramp}(s) &= \dfrac{\mu h_\kappa(s)}{\sqrt{\mu^2 h_\kappa(s)+1}} \end{aligned}\end{split}\]

Both softenings are differentiable for every finite \(s\). The narrow hinge leaves a negligible positive capacity when the circle is closed or over-driven, while preserving the current-command dependence through the transition. Increasing \(\mu\) recovers the exact current-circle leg.

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[1]

WECC REMTF, Generic Solar Photovoltaic System Dynamic Simulation Model Specification, September 2012.

[2]

The WECC Central Station Photovoltaic Power Plant Model Validation Guideline specifies that a nonpositive REEC_B mvab inherits the REGC_A base.