DCED

Type for DC-based economic dispatch.

Common Parameters: c2, c1, c0, pmax, pmin, pd, ptdf, rate_a

Common Vars: pg

Common Constraints: pb, lub, llb

Available routines: DCOPF, DCOPF2, RTED, RTEDDG, RTEDESP, RTEDES, RTEDVIS, RTED2, RTED2DG, RTED2ESP, RTED2ES, ED, EDDG, EDES, ED2, ED2DG, ED2ES, DCOPF1

DCOPF

DC optimal power flow (DCOPF) using B-theta formulation.

Notes

• The nodal price is calculated as pi in pic.

• Devices online status of StaticGen, StaticLoad, and Shunt are considered in the connectivity matrices Cft, Cg, Cl, and Csh.

References

1. R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 12-19, Feb. 2011

2. Y. Chen et al., "Security-Constrained Unit Commitment for Electricity Market: Modeling, Solution Methods, and Future Challenges," in IEEE Transactions on Power Systems, vol. 38, no. 5, pp. 4668-4681, Sept. 2023

Objective

Unit	Expression
$	\(min. \sum(c_{2} p_g^{2})+ \sum(c_{1} p_g)+ \sum(u_{g} c_{0})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin

DCOPF2

DC optimal power flow (DCOPF) using PTDF formulation.

For large cases, it is recommended to build the PTDF first, especially when incremental build is necessary.

Notes

• This routine requires PTDF matrix.

• LMP pi is calculated with two parts, energy price and congestion price.

• Bus angle aBus is calculated after solving the problem.

• In export results, pi and pic are kept for each bus, while pie can be restored manually by pie = pi - pic if needed.

Warning

In this implementation, the dual variables for constraints have opposite signs compared to the mathematical formulation: 1. The dual of pb returns a negative value, so energy price is computed as -pb.dual_variables[0]. 2. Similarly, a minus sign is applied to the duals of plfub and plflb when calculating congestion price. The reason for this sign difference is not yet fully understood.

References

1. R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 12-19, Feb. 2011

2. Y. Chen et al., "Security-Constrained Unit Commitment for Electricity Market: Modeling, Solution Methods, and Future Challenges," in IEEE Transactions on Power Systems, vol. 38, no. 5, pp. 4668-4681, Sept. 2023

Objective

Unit	Expression
$	\(min. \sum(c_{2} p_g^{2})+ \sum(c_{1} p_g)+ \sum(u_{g} c_{0})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d} - C_{sh} g_{sh} - P_{bus}^{inj})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g) - \sum(p_{d}) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

RTED

DC-based real-time economic dispatch (RTED).

RTED extends DCOPF with:

• Vars for SFR reserve: pru and prd

• Param for linear SFR cost: cru and crd

• Param for SFR requirement: du and dd

• Param for ramping: start point pg0 and ramping limit R10

• Param pg0, which can be retrieved from dynamic simulation results.

The function dc2ac sets the vBus value from solved ACOPF. Without this conversion, dynamic simulation might fail due to the gap between DC-based dispatch results and AC-based dynamic initialization.

Notes

• Formulations have been adjusted with interval config.t, 5/60 [Hour] by default.

• The tie-line flow related constraints are omitted in this formulation.

• Power generation is balanced for the entire system.

• SFR is balanced for each area.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd

Config Fields in [RTED]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTEDDG

RTED with distributed generator DG.

Note that RTEDDG only includes DG output power. If ESD1 is included, RTEDES should be used instead, otherwise there is no SOC.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
idg	\(I_{DG}\)	Index DG power from pg	VarSelect

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap

Config Fields in [RTEDDG]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTEDESP

Price run of RTED with energy storage ESD1.

This routine is not intended to work standalone. It should be used after solved RTEDES.

The binary variables ucd and udd are now parameters retrieved from solved RTEDES.

The constraints zce1 - zce3 and zde1 - zde3 are now simplified to zce and zde as below:

\[(1 - u_{cd}) * p_{ce} <= 0 (1 - u_{dd}) * p_{de} <= 0\]

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})+ T_{cfg} \sum(- c_{c,ESD} p_{c,ESD} + c_{d,ESD} p_{d,ESD})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)
cesd	Select pce and pde from pg	\(I_{ESD} p_g + p_{c,ESD} - p_{d,ESD} = 0\)
SOClb	SOC lower bound	\(-SOC + SOC_{min} \leq 0\)
SOCub	SOC upper bound	\(SOC - SOC_{max} \leq 0\)
SOCb	ESD1 SOC balance	\(E_n (SOC - SOC_{init}) - T_{cfg} \eta_c p_{c,ESD}+ T_{cfg} \frac{1}{\eta_d} p_{d,ESD} = 0\)
SOCr	ESD1 final SOC requirement	\(SOC_{end} - SOC \leq 0\)
zce	zce bound	\(1-u_{c,ESD} p_{c,ESD} \leq 0\)
zde	zde bound	\(1-u_{d,ESD} p_{d,ESD} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG
SOC	\(SOC\)	ESD1 State of Charge	p.u. (%)	ESD1	nonneg
pce	\(p_{c,ESD}\)	ESD1 charging power	p.u.	ESD1	nonneg
pde	\(p_{d,ESD}\)	ESD1 discharging power	p.u.	ESD1	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
idg	\(I_{DG}\)	Index DG power from pg	VarSelect
REtaD	\(\frac{1}{\eta_d}\)		NumOp
Mb	\(M_{big}\)	10 times of max of pmax as big M	NumOp
ies	\(I_{ESD}\)	Index ESD from StaticGen	VarSelect

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
En	\(E_n\)	Rated energy capacity	MWh	ESD1.En
SOCmax	\(SOC_{max}\)	Maximum allowed value for SOC in limiter		ESD1.SOCmax
SOCmin	\(SOC_{min}\)	Minimum required value for SOC in limiter		ESD1.SOCmin
SOCinit	\(SOC_{init}\)	Initial SOC		ESD1.SOCinit
SOCend	\(SOC_{end}\)	Target SOC at the end of the period		ESD1.SOCend
EtaC	\(\eta_c\)	Efficiency during charging		ESD1.EtaC
EtaD	\(\eta_d\)	Efficiency during discharging		ESD1.EtaD
cesdc	\(c_{c,ESD}\)	Charging cost	$/p.u.*h	ESD1.cesdc
cesdd	\(c_{d,ESD}\)	Discharging cost	$/p.u.*h	ESD1.cesdd
genesd	\(g_{ESD}\)	gen of ESD		ESD1.gen
ucd	\(u_{c,ESD}\)	Retrieved ESD1 charging decision		ESD1.ucd0
udd	\(u_{d,ESD}\)	Retrieved ESD1 discharging decision		ESD1.udd0

Config Fields in [RTEDESP]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTEDES

RTED with energy storage ESD1. The bilinear term in the formulation is linearized with big-M method.

While the formulation enforces SOCend, the ESD1 owner is not required to provide an SOC constraint for every RTED interval. The optimization treats SOCend as a terminal boundary condition, allowing the dispatcher maximum flexibility to optimize power output within the hour, provided the target is met at the interval's conclusion.

The minimum charging/discharging duration logic is implemented in tcdr and tddr. For example, the logic of tcdr is: u_{cd} >= fcd, where fcd = 1 if tdc0 > 0 and tdc > tdc0, else fcd = 0.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})+ T_{cfg} \sum(- c_{c,ESD} p_{c,ESD} + c_{d,ESD} p_{d,ESD})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
cesd	Select pce and pde from pg	\(I_{ESD} p_g + p_{c,ESD} - p_{d,ESD} = 0\)
SOClb	SOC lower bound	\(-SOC + SOC_{min} \leq 0\)
SOCub	SOC upper bound	\(SOC - SOC_{max} \leq 0\)
SOCb	ESD1 SOC balance	\(E_n (SOC - SOC_{init}) - T_{cfg} \eta_c p_{c,ESD}+ T_{cfg} \frac{1}{\eta_d} p_{d,ESD} = 0\)
SOCr	ESD1 final SOC requirement	\(SOC_{end} - SOC \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)
cdb	Charging decision bound	\(u_{c,ESD} + u_{d,ESD} - 1 = 0\)
zce1	zce bound 1	\(-z_{c,ESD} + p_{c,ESD} \leq 0\)
zce2	zce bound 2	\(z_{c,ESD} - p_{c,ESD} - M_{big} (1-u_{c,ESD}) \leq 0\)
zce3	zce bound 3	\(z_{c,ESD} - M_{big} u_{c,ESD} \leq 0\)
zde1	zde bound 1	\(-z_{d,ESD} + p_{d,ESD} \leq 0\)
zde2	zde bound 2	\(z_{d,ESD} - p_{d,ESD} - M_{big} (1-u_{d,ESD}) \leq 0\)
zde3	zde bound 3	\(z_{d,ESD} - M_{big} u_{d,ESD} \leq 0\)
tcdr	Minimum charging duration	\((t_{dc0} > 0) (t_{dc} > t_{dc0}) - u_{c,ESD} \leq 0\)
tddr	Minimum discharging duration	\((t_{dd0} > 0) (t_{dd} > t_{dd0}) - u_{d,ESD} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
SOC	\(SOC\)	ESD1 State of Charge	p.u. (%)	ESD1	nonneg
pce	\(p_{c,ESD}\)	ESD1 charging power	p.u.	ESD1	nonneg
pde	\(p_{d,ESD}\)	ESD1 discharging power	p.u.	ESD1	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG
ucd	\(u_{c,ESD}\)	ESD1 charging decision		ESD1	boolean
udd	\(u_{d,ESD}\)	ESD1 discharging decision		ESD1	boolean
zce	\(z_{c,ESD}\)	Aux var for charging, \(z_{c,ESD}=u_{c,ESD}*p_{c,ESD}\)		ESD1	nonneg
zde	\(z_{d,ESD}\)	Aux var for discharging, \(z_{d,ESD}=u_{d,ESD}*p_{d,ESD}\)		ESD1	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
REtaD	\(\frac{1}{\eta_d}\)		NumOp
Mb	\(M_{big}\)	10 times of max of pmax as big M	NumOp
ies	\(I_{ESD}\)	Index ESD from StaticGen	VarSelect
idg	\(I_{DG}\)	Index DG power from pg	VarSelect

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
En	\(E_n\)	Rated energy capacity	MWh	ESD1.En
SOCmax	\(SOC_{max}\)	Maximum allowed value for SOC in limiter		ESD1.SOCmax
SOCmin	\(SOC_{min}\)	Minimum required value for SOC in limiter		ESD1.SOCmin
SOCinit	\(SOC_{init}\)	Initial SOC		ESD1.SOCinit
SOCend	\(SOC_{end}\)	Target SOC at the end of the period		ESD1.SOCend
EtaC	\(\eta_c\)	Efficiency during charging		ESD1.EtaC
EtaD	\(\eta_d\)	Efficiency during discharging		ESD1.EtaD
cesdc	\(c_{c,ESD}\)	Charging cost	$/p.u.*h	ESD1.cesdc
cesdd	\(c_{d,ESD}\)	Discharging cost	$/p.u.*h	ESD1.cesdd
genesd	\(g_{ESD}\)	gen of ESD		ESD1.gen
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
tdc	\(t_{dc}\)	Minimum charging duration	h	ESD1.tdc
tdd	\(t_{dd}\)	Minimum discharging duration	h	ESD1.tdd
tdc0	\(t_{dc0}\)	Initial charging time	h	ESD1.tdc0
tdd0	\(t_{dd0}\)	Initial discharging time	h	ESD1.tdd0

Config Fields in [RTEDES]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTEDVIS

RTED with virtual inertia scheduling.

This class implements real-time economic dispatch with virtual inertia scheduling. Please ensure that the parameters dvm and dvd are set according to the system base.

References

1. B. She, F. Li, H. Cui, J. Wang, Q. Zhang and R. Bo, "Virtual Inertia Scheduling (VIS) for Real-Time Economic Dispatch of IBR-Penetrated Power Systems," in IEEE Transactions on Sustainable Energy, vol. 15, no. 2, pp. 938-951, April 2024, doi: 10.1109/TSTE.2023.3319307.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})+ T_{cfg} \sum(c_{m} M + c_{d} D)\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
Mub	M upper bound	\(M - M_{max} \leq 0\)
Dub	D upper bound	\(D - D_{max} \leq 0\)
Mreq	Emulated inertia requirement	\(-S_{g} M + d_{v,m} = 0\)
Dreq	Emulated damping requirement	\(-S_{g} D + d_{v,d} = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
M	\(M\)	Emulated startup time constant (M=2H)	s	VSG.M	nonneg
D	\(D\)	Emulated damping coefficient	p.u.	VSG.D	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
gvsg	\(S_{g}\)	Sum VSG vars vector in shape of area	ZonalSum

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
cm	\(c_{m}\)	Virtual inertia cost	$/s	VSGCost.cm
cd	\(c_{d}\)	Virtual damping cost	$/(p.u.)	VSGCost.cd
zvsg	\(z_{one,vsg}\)	VSG zone		VSG.zone
Mmax	\(M_{max}\)	Maximum inertia emulation	s	VSG.Mmax
Dmax	\(D_{max}\)	Maximum damping emulation	p.u.	VSG.Dmax
dvm	\(d_{v,m}\)	Emulated inertia requirement	s	VSGR.dvm
dvd	\(d_{v,d}\)	Emulated damping requirement	p.u.	VSGR.dvd

Config Fields in [RTEDVIS]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTED2

DC-based real-time economic dispatch (RTED) using PTDF.

For large cases, it is recommended to build the PTDF first, especially when incremental build is necessary.

RTED2 extends RTED with PTDF formulation:

• Uses PTDF matrix instead of B-theta for line flow calculation

• Calculates LMP with energy price and congestion price components

• Inherits all RTED features: reserves, ramping, etc.

The function dc2ac sets the vBus value from solved ACOPF. Without this conversion, dynamic simulation might fail due to the gap between DC-based dispatch results and AC-based dynamic initialization.

Notes

• Formulations have been adjusted with interval config.t, 5/60 [Hour] by default.

• The tie-line flow related constraints are omitted in this formulation.

• Power generation is balanced for the entire system.

• SFR is balanced for each area.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d} - C_{sh} g_{sh} - P_{bus}^{inj})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g) - \sum(p_{d}) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

Config Fields in [RTED2]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTED2DG

RTED with distributed generator DG using PTDF formulation.

Note that RTED2DG only includes DG output power. If ESD1 is included, RTED2ES should be used instead, otherwise there is no SOC.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d} - C_{sh} g_{sh} - P_{bus}^{inj})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g) - \sum(p_{d}) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
idg	\(I_{DG}\)	Index DG power from pg	VarSelect
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

Config Fields in [RTED2DG]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTED2ESP

Price run of RTED with energy storage ESD1 using PTDF formulation.

This routine is not intended to work standalone. It should be used after solved RTED2ES. When both are solved, RTED2ES will be used.

The binary variables ucd and udd are now parameters retrieved from solved RTED2ES.

The constraints zce1 - zce3 and zde1 - zde3 are now simplified to zce and zde as below:

\[(1 - u_{cd}) * p_{ce} <= 0 (1 - u_{dd}) * p_{de} <= 0\]

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})+ T_{cfg} \sum(- c_{c,ESD} p_{c,ESD} + c_{d,ESD} p_{d,ESD})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d} - C_{sh} g_{sh} - P_{bus}^{inj})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g) - \sum(p_{d}) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)
cesd	Select pce and pde from pg	\(I_{ESD} p_g + p_{c,ESD} - p_{d,ESD} = 0\)
SOClb	SOC lower bound	\(-SOC + SOC_{min} \leq 0\)
SOCub	SOC upper bound	\(SOC - SOC_{max} \leq 0\)
SOCb	ESD1 SOC balance	\(E_n (SOC - SOC_{init}) - T_{cfg} \eta_c p_{c,ESD}+ T_{cfg} \frac{1}{\eta_d} p_{d,ESD} = 0\)
SOCr	ESD1 final SOC requirement	\(SOC_{end} - SOC \leq 0\)
zce	zce bound	\(1-u_{c,ESD} p_{c,ESD} \leq 0\)
zde	zde bound	\(1-u_{d,ESD} p_{d,ESD} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG
SOC	\(SOC\)	ESD1 State of Charge	p.u. (%)	ESD1	nonneg
pce	\(p_{c,ESD}\)	ESD1 charging power	p.u.	ESD1	nonneg
pde	\(p_{d,ESD}\)	ESD1 discharging power	p.u.	ESD1	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
idg	\(I_{DG}\)	Index DG power from pg	VarSelect
REtaD	\(\frac{1}{\eta_d}\)		NumOp
Mb	\(M_{big}\)	10 times of max of pmax as big M	NumOp
ies	\(I_{ESD}\)	Index ESD from StaticGen	VarSelect
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
En	\(E_n\)	Rated energy capacity	MWh	ESD1.En
SOCmax	\(SOC_{max}\)	Maximum allowed value for SOC in limiter		ESD1.SOCmax
SOCmin	\(SOC_{min}\)	Minimum required value for SOC in limiter		ESD1.SOCmin
SOCinit	\(SOC_{init}\)	Initial SOC		ESD1.SOCinit
SOCend	\(SOC_{end}\)	Target SOC at the end of the period		ESD1.SOCend
EtaC	\(\eta_c\)	Efficiency during charging		ESD1.EtaC
EtaD	\(\eta_d\)	Efficiency during discharging		ESD1.EtaD
cesdc	\(c_{c,ESD}\)	Charging cost	$/p.u.*h	ESD1.cesdc
cesdd	\(c_{d,ESD}\)	Discharging cost	$/p.u.*h	ESD1.cesdd
genesd	\(g_{ESD}\)	gen of ESD		ESD1.gen
ucd	\(u_{c,ESD}\)	Retrieved ESD1 charging decision		ESD1.ucd0
udd	\(u_{d,ESD}\)	Retrieved ESD1 discharging decision		ESD1.udd0
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

Config Fields in [RTED2ESP]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

RTED2ES

RTED with energy storage ESD1. The bilinear term in the formulation is linearized with big-M method.

While the formulation enforces SOCend, the ESD1 owner is not required to provide an SOC constraint for every RTED interval. The optimization treats SOCend as a terminal boundary condition, allowing the dispatcher maximum flexibility to optimize power output within the hour, provided the target is met at the interval's conclusion.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2}) + \sum(u_{g} c_{0})+ T_{cfg} \sum(c_{1} p_g + c_{r,u} p_{r,u} + c_{r,d} p_{r,d})+ T_{cfg} \sum(- c_{c,ESD} p_{c,ESD} + c_{d,ESD} p_{d,ESD})\)

Expressions

Name	Description	Expression	Unit	Source
plf	Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d} - C_{sh} g_{sh} - P_{bus}^{inj})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g) - \sum(p_{d}) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-p_{lf} - u_{l} R_{ATEA} \leq 0\)
plfub	line flow upper bound	\(p_{lf} - u_{l} R_{ATEA} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} = 0\)
rru	RegUp reserve source	\(u_{g} (p_g + p_{r,u}) - u_{g} p_{g, max, e} \leq 0\)
rrd	RegDn reserve source	\(u_{g} (-p_g + p_{r,d}) + u_{g} p_{g, min, e} \leq 0\)
rgu	Gen ramping up	\(u_{g} (p_g-p_{g, 0}-R_{10}) \leq 0\)
rgd	Gen ramping down	\(u_{g} (-p_g+p_{g, 0}-R_{10}) \leq 0\)
cesd	Select pce and pde from pg	\(I_{ESD} p_g + p_{c,ESD} - p_{d,ESD} = 0\)
SOClb	SOC lower bound	\(-SOC + SOC_{min} \leq 0\)
SOCub	SOC upper bound	\(SOC - SOC_{max} \leq 0\)
SOCb	ESD1 SOC balance	\(E_n (SOC - SOC_{init}) - T_{cfg} \eta_c p_{c,ESD}+ T_{cfg} \frac{1}{\eta_d} p_{d,ESD} = 0\)
SOCr	ESD1 final SOC requirement	\(SOC_{end} - SOC \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)
cdb	Charging decision bound	\(u_{c,ESD} + u_{d,ESD} - 1 = 0\)
zce1	zce bound 1	\(-z_{c,ESD} + p_{c,ESD} \leq 0\)
zce2	zce bound 2	\(z_{c,ESD} - p_{c,ESD} - M_{big} (1-u_{c,ESD}) \leq 0\)
zce3	zce bound 3	\(z_{c,ESD} - M_{big} u_{c,ESD} \leq 0\)
zde1	zde bound 1	\(-z_{d,ESD} + p_{d,ESD} \leq 0\)
zde2	zde bound 2	\(z_{d,ESD} - p_{d,ESD} - M_{big} (1-u_{d,ESD}) \leq 0\)
zde3	zde bound 3	\(z_{d,ESD} - M_{big} u_{d,ESD} \leq 0\)
tcdr	Minimum charging duration	\((t_{dc0} > 0) (t_{dc} > t_{dc0}) - u_{c,ESD} \leq 0\)
tddr	Minimum discharging duration	\((t_{dd0} > 0) (t_{dd} > t_{dd0}) - u_{d,ESD} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	Gen active power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	Bus voltage magnitude, placeholder	p.u.	Bus.v
aBus	\(\theta_{bus}\)	Bus voltage angle	rad	Bus.a
pru	\(p_{r,u}\)	RegUp reserve	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	RegDn reserve	p.u.	StaticGen	nonneg
SOC	\(SOC\)	ESD1 State of Charge	p.u. (%)	ESD1	nonneg
pce	\(p_{c,ESD}\)	ESD1 charging power	p.u.	ESD1	nonneg
pde	\(p_{d,ESD}\)	ESD1 discharging power	p.u.	ESD1	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG
ucd	\(u_{c,ESD}\)	ESD1 charging decision		ESD1	boolean
udd	\(u_{d,ESD}\)	ESD1 discharging decision		ESD1	boolean
zce	\(z_{c,ESD}\)	Aux var for charging, \(z_{c,ESD}=u_{c,ESD}*p_{c,ESD}\)		ESD1	nonneg
zde	\(z_{d,ESD}\)	Aux var for discharging, \(z_{d,ESD}=u_{d,ESD}*p_{d,ESD}\)		ESD1	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
REtaD	\(\frac{1}{\eta_d}\)		NumOp
Mb	\(M_{big}\)	10 times of max of pmax as big M	NumOp
ies	\(I_{ESD}\)	Index ESD from StaticGen	VarSelect
idg	\(I_{DG}\)	Index DG power from pg	VarSelect
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
En	\(E_n\)	Rated energy capacity	MWh	ESD1.En
SOCmax	\(SOC_{max}\)	Maximum allowed value for SOC in limiter		ESD1.SOCmax
SOCmin	\(SOC_{min}\)	Minimum required value for SOC in limiter		ESD1.SOCmin
SOCinit	\(SOC_{init}\)	Initial SOC		ESD1.SOCinit
SOCend	\(SOC_{end}\)	Target SOC at the end of the period		ESD1.SOCend
EtaC	\(\eta_c\)	Efficiency during charging		ESD1.EtaC
EtaD	\(\eta_d\)	Efficiency during discharging		ESD1.EtaD
cesdc	\(c_{c,ESD}\)	Charging cost	$/p.u.*h	ESD1.cesdc
cesdd	\(c_{d,ESD}\)	Discharging cost	$/p.u.*h	ESD1.cesdd
genesd	\(g_{ESD}\)	gen of ESD		ESD1.gen
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
tdc	\(t_{dc}\)	Minimum charging duration	h	ESD1.tdc
tdd	\(t_{dd}\)	Minimum discharging duration	h	ESD1.tdd
tdc0	\(t_{dc0}\)	Initial charging time	h	ESD1.tdc0
tdd0	\(t_{dd0}\)	Initial discharging time	h	ESD1.tdd0
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

Config Fields in [RTED2ES]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	0.083	time interval in hours

ED

DC-based multi-period economic dispatch (ED). Dispatch interval config.t ($T_{cfg}$) is introduced, 1 [Hour] by default. ED extends DCOPF as follows:

• Vars pg, pru, prd are extended to 2D

• 2D Vars rgu and rgd are introduced

• Param ug is sourced from EDSlotGen.ug as generator commitment

Notes

• Formulations have been adjusted with interval config.t

• The tie-line flow is not implemented in this model.

• EDSlotGen.ug is used instead of StaticGen.u for generator commitment.

• Following reserves are balanced for each "Area": RegUp reserve rbu, RegDn reserve rbd, and Spinning reserve rsr.

Objective

Unit	Expression
$	\(min. \sum(T_{cfg}^{2} c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g + c_{sr} p_{r,s})+ \sum(u_{g} c_{0} 1_{tl})\)

Expressions

Name	Description	Expression	Unit	Source
plf	2D Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} p_{d,s} + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} 1_{tl} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} 1_{tl} = 0\)
rru	RegUp reserve source	\(p_g + p_{r,u} - u_{g} p_{g, max} 1_{tl} \leq 0\)
rrd	RegDn reserve source	\(-p_g + p_{r,d} + u_{g} p_{g, min} 1_{tl} \leq 0\)
rgu	Gen ramping up	\(p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
rgd	Gen ramping down	\(-p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
prsb	spinning reserve balance	\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
rgu0	Initial gen ramping up	\(u_{g}[: 0], p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} \leq 0\)
rgd0	Initial gen ramping down	\(u_{g}[: 0], -p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	2D Gen power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	2D Bus voltage	p.u.	Bus.v
aBus	\(\theta_{bus}\)	2D Bus angle	rad	Bus.a
pru	\(p_{r,u}\)	2D RegUp power	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	2D RegDn power	p.u.	StaticGen	nonneg
prs	\(p_{r,s}\)	spinning reserve	p.u.	StaticGen	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
tlv	\(1_{tl}\)	time length vector	NumOp
pds	\(p_{d,s}\)	Scaled load	LoadScale
Mr	\(M_{r}\)	Subtraction matrix for ramping	RampSub
RR30	\(R_{30,R}\)	Repeated ramp rate	NumHstack
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	unit commitment decisions		EDSlotGen.ug
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
timeslot	\(t_{s,idx}\)	Time slot for multi-period ED		EDSlot.idx
sd	\(s_{d}\)	area load scaling factor		EDSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr

Config Fields in [ED]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	1	time interval in hours

EDDG

ED with distributed generation DG.

Note that EDDG only includes DG output power. If ESD1 is included, EDES should be used instead, otherwise there is no SOC.

Objective

Unit	Expression
$	\(min. \sum(T_{cfg}^{2} c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g + c_{sr} p_{r,s})+ \sum(u_{g} c_{0} 1_{tl})\)

Expressions

Name	Description	Expression	Unit	Source
plf	2D Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} p_{d,s} + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} 1_{tl} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} 1_{tl} = 0\)
rru	RegUp reserve source	\(p_g + p_{r,u} - u_{g} p_{g, max} 1_{tl} \leq 0\)
rrd	RegDn reserve source	\(-p_g + p_{r,d} + u_{g} p_{g, min} 1_{tl} \leq 0\)
rgu	Gen ramping up	\(p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
rgd	Gen ramping down	\(-p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
prsb	spinning reserve balance	\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
rgu0	Initial gen ramping up	\(u_{g}[: 0], p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} \leq 0\)
rgd0	Initial gen ramping down	\(u_{g}[: 0], -p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	2D Gen power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	2D Bus voltage	p.u.	Bus.v
aBus	\(\theta_{bus}\)	2D Bus angle	rad	Bus.a
pru	\(p_{r,u}\)	2D RegUp power	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	2D RegDn power	p.u.	StaticGen	nonneg
prs	\(p_{r,s}\)	spinning reserve	p.u.	StaticGen	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
tlv	\(1_{tl}\)	time length vector	NumOp
pds	\(p_{d,s}\)	Scaled load	LoadScale
Mr	\(M_{r}\)	Subtraction matrix for ramping	RampSub
RR30	\(R_{30,R}\)	Repeated ramp rate	NumHstack
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
idg	\(I_{DG}\)	Index DG power from pg	VarSelect

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	unit commitment decisions		EDSlotGen.ug
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
timeslot	\(t_{s,idx}\)	Time slot for multi-period ED		EDSlot.idx
sd	\(s_{d}\)	area load scaling factor		EDSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap

Config Fields in [EDDG]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	1	time interval in hours

EDES

ED with energy storage ESD1. The bilinear term in the formulation is linearized with big-M method.

Objective

Unit	Expression
$	\(min. \sum(T_{cfg}^{2} c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g + c_{sr} p_{r,s})+ \sum(u_{g} c_{0} 1_{tl})+ T_{cfg} \sum(- c_{c,ESD} p_{c,ESD} + c_{d,ESD} p_{d,ESD})\)

Expressions

Name	Description	Expression	Unit	Source
plf	2D Line flow	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl}\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} p_{d,s} + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} 1_{tl} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} 1_{tl} = 0\)
rru	RegUp reserve source	\(p_g + p_{r,u} - u_{g} p_{g, max} 1_{tl} \leq 0\)
rrd	RegDn reserve source	\(-p_g + p_{r,d} + u_{g} p_{g, min} 1_{tl} \leq 0\)
rgu	Gen ramping up	\(p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
rgd	Gen ramping down	\(-p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
prsb	spinning reserve balance	\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
rgu0	Initial gen ramping up	\(u_{g}[: 0], p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} \leq 0\)
rgd0	Initial gen ramping down	\(u_{g}[: 0], -p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} \leq 0\)
cesd	Select pce and pde from pg	\(I_{ESD} p_g + p_{c,ESD} - p_{d,ESD} = 0\)
SOClb	SOC lower bound	\(-SOC + SOC_{min} \leq 0\)
SOCub	SOC upper bound	\(SOC - SOC_{max} \leq 0\)
SOCb	ESD1 SOC balance	\(E_{n,R} SOC M_{r,ES} - T_{cfg} \eta_{c,R} p_{c,ESD}[:, 1:] + T_{cfg} R_{\eta_d,R} p_{d,ESD}[:, 1:] = 0\)
SOCr	ESD1 final SOC requirement	\(SOC_{end} - SOC[:, -1] \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)
cdb	Charging decision bound	\(u_{c,ESD} + u_{d,ESD} - 1 = 0\)
zce1	zce bound 1	\(-z_{c,ESD} + p_{c,ESD} \leq 0\)
zce2	zce bound 2	\(z_{c,ESD} - p_{c,ESD} - M_{big} (1-u_{c,ESD}) \leq 0\)
zce3	zce bound 3	\(z_{c,ESD} - M_{big} u_{c,ESD} \leq 0\)
zde1	zde bound 1	\(-z_{d,ESD} + p_{d,ESD} \leq 0\)
zde2	zde bound 2	\(z_{d,ESD} - p_{d,ESD} - M_{big} (1-u_{d,ESD}) \leq 0\)
zde3	zde bound 3	\(z_{d,ESD} - M_{big} u_{d,ESD} \leq 0\)
tcdr	Minimum charging duration	\((t_{dc0} > 0) (t_{dc} > t_{dc0}) - u_{c,ESD} \leq 0\)
tddr	Minimum discharging duration	\((t_{dd0} > 0) (t_{dd} > t_{dd0}) - u_{d,ESD} \leq 0\)
SOCb0	ESD1 SOC initial balance	\(E_n SOC[:, 0] - SOC_{init} - T_{cfg} \eta_c p_{c,ESD}[:, 0] + T_{cfg} \frac{1}{\eta_d} p_{d,ESD}[:, 0] = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	2D Gen power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	2D Bus voltage	p.u.	Bus.v
aBus	\(\theta_{bus}\)	2D Bus angle	rad	Bus.a
pru	\(p_{r,u}\)	2D RegUp power	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	2D RegDn power	p.u.	StaticGen	nonneg
prs	\(p_{r,s}\)	spinning reserve	p.u.	StaticGen	nonneg
SOC	\(SOC\)	ESD1 State of Charge	p.u. (%)	ESD1	nonneg
pce	\(p_{c,ESD}\)	ESD1 charging power	p.u.	ESD1	nonneg
pde	\(p_{d,ESD}\)	ESD1 discharging power	p.u.	ESD1	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG
ucd	\(u_{c,ESD}\)	ESD1 charging decision		ESD1	boolean
udd	\(u_{d,ESD}\)	ESD1 discharging decision		ESD1	boolean
zce	\(z_{c,ESD}\)	Aux var for charging, \(z_{c,ESD}=u_{c,ESD}*p_{c,ESD}\)		ESD1	nonneg
zde	\(z_{d,ESD}\)	Aux var for discharging, \(z_{d,ESD}=u_{d,ESD}*p_{d,ESD}\)		ESD1	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	LMP, dual of <pb>	\(\phi[pb]\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
tlv	\(1_{tl}\)	time length vector	NumOp
pds	\(p_{d,s}\)	Scaled load	LoadScale
Mr	\(M_{r}\)	Subtraction matrix for ramping	RampSub
RR30	\(R_{30,R}\)	Repeated ramp rate	NumHstack
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
REtaD	\(\frac{1}{\eta_d}\)		NumOp
Mb	\(M_{big}\)	10 times of max of pmax as big M	NumOp
ies	\(I_{ESD}\)	Index ESD from StaticGen	VarSelect
idg	\(I_{DG}\)	Index DG power from pg	VarSelect
Mre	\(M_{r,ES}\)	Subtraction matrix for SOC	RampSub
EnR	\(E_{n,R}\)	Repeated En as 2D matrix, (ng, ng-1)	NumHstack
EtaCR	\(\eta_{c,R}\)	Repeated Etac as 2D matrix, (ng, ng-1)	NumHstack
REtaDR	\(R_{\eta_d,R}\)	Repeated REtaD as 2D matrix, (ng, ng-1)	NumHstack

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	unit commitment decisions		EDSlotGen.ug
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
timeslot	\(t_{s,idx}\)	Time slot for multi-period ED		EDSlot.idx
sd	\(s_{d}\)	area load scaling factor		EDSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
En	\(E_n\)	Rated energy capacity	MWh	ESD1.En
SOCmax	\(SOC_{max}\)	Maximum allowed value for SOC in limiter		ESD1.SOCmax
SOCmin	\(SOC_{min}\)	Minimum required value for SOC in limiter		ESD1.SOCmin
SOCinit	\(SOC_{init}\)	Initial SOC		ESD1.SOCinit
SOCend	\(SOC_{end}\)	Target SOC at the end of the period		ESD1.SOCend
EtaC	\(\eta_c\)	Efficiency during charging		ESD1.EtaC
EtaD	\(\eta_d\)	Efficiency during discharging		ESD1.EtaD
cesdc	\(c_{c,ESD}\)	Charging cost	$/p.u.*h	ESD1.cesdc
cesdd	\(c_{d,ESD}\)	Discharging cost	$/p.u.*h	ESD1.cesdd
genesd	\(g_{ESD}\)	gen of ESD		ESD1.gen
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
tdc	\(t_{dc}\)	Minimum charging duration	h	ESD1.tdc
tdd	\(t_{dd}\)	Minimum discharging duration	h	ESD1.tdd
tdc0	\(t_{dc0}\)	Initial charging time	h	ESD1.tdc0
tdd0	\(t_{dd0}\)	Initial discharging time	h	ESD1.tdd0

Config Fields in [EDES]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	1	time interval in hours

ED2

DC-based multi-period economic dispatch (ED) using PTDF. Dispatch interval config.t ($T_{cfg}$) is introduced, 1 [Hour] by default. ED extends DCOPF as follows:

• Vars pg, pru, prd are extended to 2D

• 2D Vars rgu and rgd are introduced

• Param ug is sourced from EDSlotGen.ug as generator commitment

Notes

• Formulations have been adjusted with interval config.t

• The tie-line flow is not implemented in this model.

• EDSlotGen.ug is used instead of StaticGen.u for generator commitment.

• Following reserves are balanced for each "Area": RegUp reserve rbu, RegDn reserve rbd, and Spinning reserve rsr.

Objective

Unit	Expression
$	\(min. \sum(T_{cfg}^{2} c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g + c_{sr} p_{r,s})+ \sum(u_{g} c_{0} 1_{tl})\)

Expressions

Name	Description	Expression	Unit	Source
plf	2D Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d,s} - C_{sh} g_{sh} 1_{tl} - P_{bus}^{inj} 1_{tl})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g, axis=0) - \sum(p_{d,s}, axis=0) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} 1_{tl} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} 1_{tl} = 0\)
rru	RegUp reserve source	\(p_g + p_{r,u} - u_{g} p_{g, max} 1_{tl} \leq 0\)
rrd	RegDn reserve source	\(-p_g + p_{r,d} + u_{g} p_{g, min} 1_{tl} \leq 0\)
rgu	Gen ramping up	\(p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
rgd	Gen ramping down	\(-p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
prsb	spinning reserve balance	\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
rgu0	Initial gen ramping up	\(u_{g}[: 0], p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} \leq 0\)
rgd0	Initial gen ramping down	\(u_{g}[: 0], -p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} \leq 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	2D Gen power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	2D Bus voltage	p.u.	Bus.v
aBus	\(\theta_{bus}\)	2D Bus angle	rad	Bus.a
pru	\(p_{r,u}\)	2D RegUp power	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	2D RegDn power	p.u.	StaticGen	nonneg
prs	\(p_{r,s}\)	spinning reserve	p.u.	StaticGen	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
tlv	\(1_{tl}\)	time length vector	NumOp
pds	\(p_{d,s}\)	Scaled load	LoadScale
Mr	\(M_{r}\)	Subtraction matrix for ramping	RampSub
RR30	\(R_{30,R}\)	Repeated ramp rate	NumHstack
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	unit commitment decisions		EDSlotGen.ug
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
timeslot	\(t_{s,idx}\)	Time slot for multi-period ED		EDSlot.idx
sd	\(s_{d}\)	area load scaling factor		EDSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

Config Fields in [ED2]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	1	time interval in hours

ED2DG

ED with distributed generation DG using PTDF.

Note that ED2DG only includes DG output power. If ESD1 is included, ED2ES should be used instead, otherwise there is no SOC.

Objective

Unit	Expression
$	\(min. \sum(T_{cfg}^{2} c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g + c_{sr} p_{r,s})+ \sum(u_{g} c_{0} 1_{tl})\)

Expressions

Name	Description	Expression	Unit	Source
plf	2D Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d,s} - C_{sh} g_{sh} 1_{tl} - P_{bus}^{inj} 1_{tl})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g, axis=0) - \sum(p_{d,s}, axis=0) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} 1_{tl} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} 1_{tl} = 0\)
rru	RegUp reserve source	\(p_g + p_{r,u} - u_{g} p_{g, max} 1_{tl} \leq 0\)
rrd	RegDn reserve source	\(-p_g + p_{r,d} + u_{g} p_{g, min} 1_{tl} \leq 0\)
rgu	Gen ramping up	\(p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
rgd	Gen ramping down	\(-p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
prsb	spinning reserve balance	\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
rgu0	Initial gen ramping up	\(u_{g}[: 0], p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} \leq 0\)
rgd0	Initial gen ramping down	\(u_{g}[: 0], -p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	2D Gen power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	2D Bus voltage	p.u.	Bus.v
aBus	\(\theta_{bus}\)	2D Bus angle	rad	Bus.a
pru	\(p_{r,u}\)	2D RegUp power	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	2D RegDn power	p.u.	StaticGen	nonneg
prs	\(p_{r,s}\)	spinning reserve	p.u.	StaticGen	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
tlv	\(1_{tl}\)	time length vector	NumOp
pds	\(p_{d,s}\)	Scaled load	LoadScale
Mr	\(M_{r}\)	Subtraction matrix for ramping	RampSub
RR30	\(R_{30,R}\)	Repeated ramp rate	NumHstack
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp
idg	\(I_{DG}\)	Index DG power from pg	VarSelect

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	unit commitment decisions		EDSlotGen.ug
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
timeslot	\(t_{s,idx}\)	Time slot for multi-period ED		EDSlot.idx
sd	\(s_{d}\)	area load scaling factor		EDSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap

Config Fields in [ED2DG]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	1	time interval in hours

ED2ES

ED with energy storage ESD1 using PTDF.

Objective

Unit	Expression
$	\(min. \sum(T_{cfg}^{2} c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g + c_{sr} p_{r,s})+ \sum(u_{g} c_{0} 1_{tl})+ T_{cfg} \sum(- c_{c,ESD} p_{c,ESD} + c_{d,ESD} p_{d,ESD})\)

Expressions

Name	Description	Expression	Unit	Source
plf	2D Line flow	\(P_{TDF} (C_{g} p_g - C_{l} p_{d,s} - C_{sh} g_{sh} 1_{tl} - P_{bus}^{inj} 1_{tl})\)	p.u.	Line
pmaxe	Effective pmax	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, max}\)	p.u.	StaticGen
pmine	Effective pmin	\(c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min}\)	p.u.	StaticGen

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g, axis=0) - \sum(p_{d,s}, axis=0) = 0\)
sbus	align slack bus angle	\(I_{sb} \theta_{bus} = 0\)
pglb	pg min	\(-p_g + p_{g, min, e} \leq 0\)
pgub	pg max	\(p_g - p_{g, max, e} \leq 0\)
plflb	line flow lower bound	\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - u_{l} R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} + \theta_{bus, min} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
rbu	RegUp reserve balance	\(S_{g} u_{g} p_{r,u} - d_{u, d} 1_{tl} = 0\)
rbd	RegDn reserve balance	\(S_{g} u_{g} p_{r,d} - d_{d, d} 1_{tl} = 0\)
rru	RegUp reserve source	\(p_g + p_{r,u} - u_{g} p_{g, max} 1_{tl} \leq 0\)
rrd	RegDn reserve source	\(-p_g + p_{r,d} + u_{g} p_{g, min} 1_{tl} \leq 0\)
rgu	Gen ramping up	\(p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
rgd	Gen ramping down	\(-p_g M_{r} - T_{cfg} R_{30,R} \leq 0\)
prsb	spinning reserve balance	\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
rgu0	Initial gen ramping up	\(u_{g}[: 0], p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} \leq 0\)
rgd0	Initial gen ramping down	\(u_{g}[: 0], -p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} \leq 0\)
cesd	Select pce and pde from pg	\(I_{ESD} p_g + p_{c,ESD} - p_{d,ESD} = 0\)
SOClb	SOC lower bound	\(-SOC + SOC_{min} \leq 0\)
SOCub	SOC upper bound	\(SOC - SOC_{max} \leq 0\)
SOCb	ESD1 SOC balance	\(E_{n,R} SOC M_{r,ES} - T_{cfg} \eta_{c,R} p_{c,ESD}[:, 1:] + T_{cfg} R_{\eta_d,R} p_{d,ESD}[:, 1:] = 0\)
SOCr	ESD1 final SOC requirement	\(SOC_{end} - SOC[:, -1] \leq 0\)
cdgb	Select DG power from pg	\(I_{DG} p_g - p_{g,DG} = 0\)
cdb	Charging decision bound	\(u_{c,ESD} + u_{d,ESD} - 1 = 0\)
zce1	zce bound 1	\(-z_{c,ESD} + p_{c,ESD} \leq 0\)
zce2	zce bound 2	\(z_{c,ESD} - p_{c,ESD} - M_{big} (1-u_{c,ESD}) \leq 0\)
zce3	zce bound 3	\(z_{c,ESD} - M_{big} u_{c,ESD} \leq 0\)
zde1	zde bound 1	\(-z_{d,ESD} + p_{d,ESD} \leq 0\)
zde2	zde bound 2	\(z_{d,ESD} - p_{d,ESD} - M_{big} (1-u_{d,ESD}) \leq 0\)
zde3	zde bound 3	\(z_{d,ESD} - M_{big} u_{d,ESD} \leq 0\)
tcdr	Minimum charging duration	\((t_{dc0} > 0) (t_{dc} > t_{dc0}) - u_{c,ESD} \leq 0\)
tddr	Minimum discharging duration	\((t_{dd0} > 0) (t_{dd} > t_{dd0}) - u_{d,ESD} \leq 0\)
SOCb0	ESD1 SOC initial balance	\(E_n SOC[:, 0] - SOC_{init} - T_{cfg} \eta_c p_{c,ESD}[:, 0] + T_{cfg} \frac{1}{\eta_d} p_{d,ESD}[:, 0] = 0\)

Vars

Name	Symbol	Description	Unit	Source	Properties
pg	\(p_g\)	2D Gen power	p.u.	StaticGen.p
vBus	\(v_{Bus}\)	2D Bus voltage	p.u.	Bus.v
aBus	\(\theta_{bus}\)	2D Bus angle	rad	Bus.a
pru	\(p_{r,u}\)	2D RegUp power	p.u.	StaticGen	nonneg
prd	\(p_{r,d}\)	2D RegDn power	p.u.	StaticGen	nonneg
prs	\(p_{r,s}\)	spinning reserve	p.u.	StaticGen	nonneg
SOC	\(SOC\)	ESD1 State of Charge	p.u. (%)	ESD1	nonneg
pce	\(p_{c,ESD}\)	ESD1 charging power	p.u.	ESD1	nonneg
pde	\(p_{d,ESD}\)	ESD1 discharging power	p.u.	ESD1	nonneg
pgdg	\(p_{g,DG}\)	DG output power	p.u.	DG
ucd	\(u_{c,ESD}\)	ESD1 charging decision		ESD1	boolean
udd	\(u_{d,ESD}\)	ESD1 discharging decision		ESD1	boolean
zce	\(z_{c,ESD}\)	Aux var for charging, \(z_{c,ESD}=u_{c,ESD}*p_{c,ESD}\)		ESD1	nonneg
zde	\(z_{d,ESD}\)	Aux var for discharging, \(z_{d,ESD}=u_{d,ESD}*p_{d,ESD}\)		ESD1	nonneg

ExpressionCalcs

Name	Description	Expression	Unit	Source
pi	locational marginal price (LMP)	\(-\phi[pb] - P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus
mu1	Lagrange multipliers, dual of <plflb>	\(\phi[plflb]\)	$/p.u.	Line
mu2	Lagrange multipliers, dual of <plfub>	\(\phi[plfub]\)	$/p.u.	Line
pie	Energy price	\(-\phi[pb]\)	$/p.u.
pic	Congestion price	\(-P_{TDF}^T (\phi[plfub] - \phi[plflb])\)	$/p.u.	Bus

Services

Name	Symbol	Description	Type
pd	\(p_{d}\)	effective active demand	NumOpDual
isb	\(I_{sb}\)	Index slack bus from all buses	VarSelect
ctrle	\(c_{trl, e}\)	Effective Gen controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Effective Gen uncontrollability	NumOpDual
gs	\(S_{g}\)	Sum Gen vars vector in shape of area	ZonalSum
ds	\(S_{d}\)	Sum pd vector in shape of area	ZonalSum
pdz	\(p_{d,z}\)	zonal total load	NumOpDual
dud	\(d_{u, d}\)	zonal RegUp reserve requirement	NumOpDual
ddd	\(d_{d, d}\)	zonal RegDn reserve requirement	NumOpDual
tlv	\(1_{tl}\)	time length vector	NumOp
pds	\(p_{d,s}\)	Scaled load	LoadScale
Mr	\(M_{r}\)	Subtraction matrix for ramping	RampSub
RR30	\(R_{30,R}\)	Repeated ramp rate	NumHstack
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp
REtaD	\(\frac{1}{\eta_d}\)		NumOp
Mb	\(M_{big}\)	10 times of max of pmax as big M	NumOp
ies	\(I_{ESD}\)	Index ESD from StaticGen	VarSelect
idg	\(I_{DG}\)	Index DG power from pg	VarSelect
Mre	\(M_{r,ES}\)	Subtraction matrix for SOC	RampSub
EnR	\(E_{n,R}\)	Repeated En as 2D matrix, (ng, ng-1)	NumHstack
EtaCR	\(\eta_{c,R}\)	Repeated Etac as 2D matrix, (ng, ng-1)	NumHstack
REtaDR	\(R_{\eta_d,R}\)	Repeated REtaD as 2D matrix, (ng, ng-1)	NumHstack

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	unit commitment decisions		EDSlotGen.ug
pg0	\(p_{g, 0}\)	Gen initial active power	p.u.	StaticGen.p0
gsh	\(g_{sh}\)	shunt conductance		Shunt.g
buss	\(B_{us,s}\)	Bus slack		Slack.bus
ud	\(u_{d}\)	Load connection status		StaticLoad.u
pd0	\(p_{d,0}\)	active demand	p.u.	StaticLoad.p0
Cg	\(C_{g}\)	Gen connection matrix		MatProcessor.Cg
Cl	\(C_{l}\)	Load connection matrix		MatProcessor.Cl
CftT	\(C_{ft}^T\)	Transpose of line connection matrix		MatProcessor.CftT
Csh	\(C_{sh}\)	Shunt connection matrix		MatProcessor.Csh
Bbus	\(B_{bus}\)	Bus admittance matrix		MatProcessor.Bbus
Bf	\(B_{f}\)	Bf matrix		MatProcessor.Bf
Pbusinj	\(P_{bus}^{inj}\)	Bus power injection vector		MatProcessor.Pbusinj
Pfinj	\(P_{f}^{inj}\)	Line power injection vector		MatProcessor.Pfinj
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
ctrl	\(c_{trl}\)	Gen controllability		StaticGen.ctrl
pmax	\(p_{g, max}\)	Gen maximum active power	p.u.	StaticGen.pmax
pmin	\(p_{g, min}\)	Gen minimum active power	p.u.	StaticGen.pmin
ul	\(u_{l}\)	Line connection status		Line.u
rate_a	\(R_{ATEA}\)	long-term flow limit	p.u.	Line.rate_a
amax	\(\theta_{bus, max}\)	max line angle difference		Line.amax
amin	\(\theta_{bus, min}\)	min line angle difference		Line.amin
zg	\(z_{one,g}\)	Gen area		StaticGen.area
zd	\(z_{one,d}\)	Load area		StaticLoad.area
R10	\(R_{10}\)	10-min ramp rate	p.u./h	StaticGen.R10
cru	\(c_{r,u}\)	RegUp reserve coefficient	$/(p.u.)	SFRCost.cru
crd	\(c_{r,d}\)	RegDown reserve coefficient	$/(p.u.)	SFRCost.crd
du	\(d_{u}\)	RegUp reserve requirement in percentage	%	SFR.du
dd	\(d_{d}\)	RegDown reserve requirement in percentage	%	SFR.dd
timeslot	\(t_{s,idx}\)	Time slot for multi-period ED		EDSlot.idx
sd	\(s_{d}\)	area load scaling factor		EDSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF
En	\(E_n\)	Rated energy capacity	MWh	ESD1.En
SOCmax	\(SOC_{max}\)	Maximum allowed value for SOC in limiter		ESD1.SOCmax
SOCmin	\(SOC_{min}\)	Minimum required value for SOC in limiter		ESD1.SOCmin
SOCinit	\(SOC_{init}\)	Initial SOC		ESD1.SOCinit
SOCend	\(SOC_{end}\)	Target SOC at the end of the period		ESD1.SOCend
EtaC	\(\eta_c\)	Efficiency during charging		ESD1.EtaC
EtaD	\(\eta_d\)	Efficiency during discharging		ESD1.EtaD
cesdc	\(c_{c,ESD}\)	Charging cost	$/p.u.*h	ESD1.cesdc
cesdd	\(c_{d,ESD}\)	Discharging cost	$/p.u.*h	ESD1.cesdd
genesd	\(g_{ESD}\)	gen of ESD		ESD1.gen
gendg	\(g_{DG}\)	gen of DG		DG.gen
gammapd	\(\gamma_{p,DG}\)	Ratio of DG.pge w.r.t to that of static generator		DG.gammap
tdc	\(t_{dc}\)	Minimum charging duration	h	ESD1.tdc
tdd	\(t_{dd}\)	Minimum discharging duration	h	ESD1.tdd
tdc0	\(t_{dc0}\)	Initial charging time	h	ESD1.tdc0
tdd0	\(t_{dd0}\)	Initial discharging time	h	ESD1.tdd0

Config Fields in [ED2ES]

Option	Symbol	Value	Info	Accepted values
t	\(T_{cfg}\)	1	time interval in hours

DCOPF1

DC optimal power flow using PYPOWER.

This routine provides a wrapper for running DC optimal power flow analysis using the PYPOWER. It leverages PYPOWER's internal DC optimal power flow solver and maps results back to the AMS system.

In PYPOWER, the c0 term (the constant coefficient in the generator cost function) is always included in the objective, regardless of the generator's commitment status. See pypower/opf_costfcn.py for implementation details.

Notes

• This class does not implement the AMS-style DC optimal power flow formulation.

• For detailed mathematical formulations and algorithmic details, refer to the MATPOWER User's Manual, section on Optimal Power Flow.

• Algorithms 400, 500, 600, and 700 are not fully supported yet.

Added in version 1.0.10.

Objective

Unit	Expression
$	\(min. \sum(c_{2} p_{g}^{2}) + \sum(c_{1} p_{g}) + \sum(u_{g} c_{0})\)

Vars

Name	Symbol	Description	Unit	Source	Properties
aBus	\(a_{Bus}\)	bus voltage angle	rad	Bus.a
vBus	\(v_{Bus}\)	Bus voltage magnitude	p.u.	Bus.v
pg	\(p_{g}\)	Gen active power	p.u.	StaticGen.p
qg	\(q_{g}\)	Gen reactive power	p.u.	StaticGen.q
plf	\(p_{lf}\)	Line flow	p.u.	Line
pi	\(pi\)	Lagrange multiplier on real power mismatch	$/p.u.	Bus
piq	\(piq\)	Lagrange multiplier on reactive power mismatch	$/p.u.	Bus
mu1	\(mu1\)	Kuhn-Tucker multiplier on MVA limit at bus1	$/p.u.	Line
mu2	\(mu2\)	Kuhn-Tucker multiplier on MVA limit at bus2	$/p.u.	Line

ExpressionCalcs

Name	Description	Expression	Unit	Source
None	Total cost	\(\sum(c_{2} p_{g}^{2})+ \sum(c_{1} p_{g})+ \sum(u_{g} c_{0})\)	$

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	Gen connection status		StaticGen.u
c2	\(c_{2}\)	Gen cost coefficient 2	$/(p.u.^2)	GCost.c2
c1	\(c_{1}\)	Gen cost coefficient 1	$/(p.u.)	GCost.c1
c0	\(c_{0}\)	Gen cost coefficient 0	$	GCost.c0
pd	\(p_{d}\)	active demand	p.u.	StaticLoad.p0
qd	\(q_{d}\)	reactive demand	p.u.	StaticLoad.q0

Config Fields in [DCOPF1]

Option	Symbol	Value	Info	Accepted values
verbose	\(v_{erbose}\)	1	0: no progress info, 1: little, 2: lots, 3: all	(0, 1, 2, 3)
out_all	\(o_{ut\_all}\)	0	-1: individual flags control what prints, 0: none, 1: all	(-1, 0, 1)
out_sys_sum	\(o_{ut\_sys\_sum}\)	1	print system summary	(0, 1)
out_area_sum	\(o_{ut\_area\_sum}\)	0	print area summaries	(0, 1)
out_bus	\(o_{ut\_bus}\)	1	print bus detail	(0, 1)
out_branch	\(o_{ut\_branch}\)	1	print branch detail	(0, 1)
out_gen	\(o_{ut\_gen}\)	0	print generator detail (OUT_BUS also includes gen info)	(0, 1)
out_all_lim	\(o_{ut\_all\_lim}\)	-1	-1: individual flags, 0: none, 1: binding, 2: all	(-1, 0, 1, 2)
out_v_lim	\(o_{ut\_v\_lim}\)	1	0: don't print, 1: binding constraints only, 2: all constraints	(0, 1, 2)
out_line_lim	\(o_{ut\_line\_lim}\)	1	0: don't print, 1: binding constraints only, 2: all constraints	(0, 1, 2)
out_pg_lim	\(o_{ut\_pg\_lim}\)	1	0: don't print, 1: binding constraints only, 2: all constraints	(0, 1, 2)
out_qg_lim	\(o_{ut\_qg\_lim}\)	1	0: don't print, 1: binding constraints only, 2: all constraints	(0, 1, 2)
opf_alg_dc	\(o_{pf\_alg\_dc}\)	200	0: choose default solver based on availability, 200: PIPS, 250: PIPS-sc, 400: IPOPT, 500: CPLEX, 600: MOSEK, 700: GUROBI	(0, 200, 250, 400, 500, 600, 700)
opf_violation	\(o_{pf\_violation}\)	0.000	constraint violation tolerance	>=0
opf_flow_lim	\(o_{pf\_flow\_lim}\)	0	qty to limit for branch flow constraints: 0 - apparent power flow (limit in MVA), 1 - active power flow (limit in MW), 2 - current magnitude (limit in MVA at 1 p.u. voltage)	(0, 1, 2)
opf_ignore_ang_lim	\(o_{pf\_ignore\_ang\_lim}\)	0	ignore angle difference limits for branches even if specified	(0, 1)
grb_method	\(o_{grb\_method}\)	1	0 - primal simplex, 1 - dual simplex, 2 - barrier, 3 - concurrent (LP only), 4 - deterministic concurrent (LP only)	(0, 1, 2, 3, 4)
grb_timelimit	\(o_{grb\_timelimit}\)	inf	maximum time allowed for solver (TimeLimit)	(0, inf)
grb_threads	\(o_{grb\_threads}\)	0	(auto) maximum number of threads to use (Threads)	(0, 1)
grb_opt	\(o_{grb\_opt}\)	0	See gurobi_options() for details	(0, 1)
pdipm_feastol	\(o_{pdipm\_feastol}\)	0	feasibility (equality) tolerance for Primal-Dual Interior Points Methods, set to value of OPF_VIOLATION by default	>=0
pdipm_gradtol	\(o_{pdipm\_gradtol}\)	0.000	gradient tolerance for Primal-Dual Interior Points Methods	>=0
pdipm_comptol	\(o_{pdipm\_comptol}\)	0.000	complementary condition (inequality) tolerance for Primal-Dual Interior Points Methods	>=0
pdipm_costtol	\(o_{pdipm\_costtol}\)	0.000	optimality tolerance for Primal-Dual Interior Points Methods	>=0
pdipm_max_it	\(o_{pdipm\_max\_it}\)	150	maximum iterations for Primal-Dual Interior Points Methods	>=0
scpdipm_red_it	\(o_{scpdipm\_red\_it}\)	20	maximum reductions per iteration for Step-Control Primal-Dual Interior Points Methods	>=0