DCUC

Type for DC-based unit commitment.

Available routines: UC, UCDG, UCES, UC2, UC2DG, UC2ES

UC

DC-based unit commitment (UC): The bilinear term in the formulation is linearized with big-M method.

Non-negative var pdu is introduced as unserved load with its penalty cdp.

Constraints include power balance, ramping, spinning reserve, non-spinning reserve, minimum ON/OFF duration. The cost inludes generation cost, startup cost, shutdown cost, spinning reserve cost, non-spinning reserve cost, and unserved load penalty.

Method _initial_guess is used to make initial guess for commitment decision if all generators are online at initial. It is a simple heuristic method, which may not be optimal.

Notes

• The formulations have been adjusted with interval config.t

• The tie-line flow has not been implemented in formulations.

References

1. Huang, Y., Pardalos, P. M., & Zheng, Q. P. (2017). Electrical power unit commitment: deterministic and two-stage stochastic programming models and algorithms. Springer.

2. D. A. Tejada-Arango, S. Lumbreras, P. Sánchez-Martín and A. Ramos, "Which Unit-Commitment Formulation is Best? A Comparison Framework," in IEEE Transactions on Power Systems, vol. 35, no. 4, pp. 2926-2936, July 2020, doi: 10.1109/TPWRS.2019.2962024.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g)+ \sum(u_{g} c_{0} 1_{tl})+ \sum(c_{su} v_{g,d} + c_{sd} w_{g,d}) + T_{cfg} \sum(c_{sr} p_{r,s} + c_{nsr} p_{r, ns} + c_{d,p} p_{d,u})\)

Expressions

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

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} (p_{d,s}-p_{d,u}) + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\)
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	\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
prnsb	non-spinning reserve balance	\(1-u_{g,d} p_{g, max} 1_{tl} - p_{r, ns} = 0\)
rnsr	non-spinning reserve requirement	\(-S_{g} p_{r, ns} + d_{nsr} \leq 0\)
prsb	spinning reserve balance	\(u_{g,d} p_{g, max} 1_{tl} - z_{u_{g}} - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
actv	startup action	\(u_{g,d} M_{r} - v_{g,d}[:, 1:] = 0\)
actv0	initial startup action	\(u_{g,d}[:, 0] - u_{g}[:, 0] - v_{g,d}[:, 0] = 0\)
actw	shutdown action	\(-u_{g,d} M_{r} - w_{g,d}[:, 1:] = 0\)
actw0	initial shutdown action	\(-u_{g,d}[:, 0] + u_{g}[:, 0] - w_{g,d}[:, 0] = 0\)
zuglb	zug lower bound	\(- z_{u_{g}} + p_g \leq 0\)
zugub	zug upper bound	\(z_{u_{g}} - p_g - M_{zug} (1 - u_{g,d}) \leq 0\)
zugub2	zug upper bound	\(z_{u_{g}} - M_{zug} u_{g,d} \leq 0\)
don	minimum online duration	\(T_{on} v_{g,d} - u_{g,d} \leq 0\)
doff	minimum offline duration	\(T_{off} w_{g,d} - (1 - u_{g,d}) \leq 0\)
pdumax	unserved demand upper bound	\(p_{d,u} - p_{d,s}^{+} c_{trl,d} 1_{tl} \leq 0\)

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
prns	\(p_{r, ns}\)	2D Non-spinning reserve		StaticGen	nonneg
prs	\(p_{r,s}\)	2D Spinning reserve	p.u.	StaticGen	nonneg
ugd	\(u_{g,d}\)	commitment decision		StaticGen.u	boolean
vgd	\(v_{g,d}\)	startup action		StaticGen.u	boolean
wgd	\(w_{g,d}\)	shutdown action		StaticGen.u	boolean
zug	\(z_{ug}\)	Aux var, \(z_{ug} = u_{g,d} * p_g\)		StaticGen	pos
pdu	\(p_{d,u}\)	unserved demand	p.u.	StaticLoad	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	\(c_{trl, e}\)	Reshaped controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Reshaped non-controllability	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
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
dnsrpz	\(d_{nsr, p, z}\)	zonal non-spinning reserve requirement in percentage	NumOpDual
dnsr	\(d_{nsr}\)	zonal non-spinning reserve requirement	NumOpDual
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
Mzug	\(M_{zug}\)	10 times of max of pmax as big M for zug	NumOp
Con	\(T_{on}\)	minimum ON coefficient	MinDur
Coff	\(T_{off}\)	minimum OFF coefficient	MinDur
pdsp	\(p_{d,s}^{+}\)	positive demand	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
timeslot	\(t_{s,idx}\)	Time slot for multi-period UC		UCSlot.idx
sd	\(s_{d}\)	area load scaling factor for UC		UCSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
cnsr	\(c_{nsr}\)	cost for non-spinning reserve	$/(p.u.*h)	NSRCost.cnsr
dnsr	\(d_{nsr}\)	non-spinning reserve requirement in percentage	%	NSR.demand
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
csu	\(c_{su}\)	startup cost	$	GCost.csu
csd	\(c_{sd}\)	shutdown cost	$	GCost.csd
cdp	\(c_{d,p}\)	penalty for unserved load	$/(p.u.*h)	DCost.cdp
dctrl	\(c_{trl,d}\)	load controllability		StaticLoad.ctrl
td1	\(t_{d1}\)	minimum ON duration	h	StaticGen.td1
td2	\(t_{d2}\)	minimum OFF duration	h	StaticGen.td2

Config Fields in [UC]

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

UCDG

UC with distributed generation DG.

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

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g)+ \sum(u_{g} c_{0} 1_{tl})+ \sum(c_{su} v_{g,d} + c_{sd} w_{g,d}) + T_{cfg} \sum(c_{sr} p_{r,s} + c_{nsr} p_{r, ns} + c_{d,p} p_{d,u})\)

Expressions

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

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} (p_{d,s}-p_{d,u}) + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\)
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	\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
prnsb	non-spinning reserve balance	\(1-u_{g,d} p_{g, max} 1_{tl} - p_{r, ns} = 0\)
rnsr	non-spinning reserve requirement	\(-S_{g} p_{r, ns} + d_{nsr} \leq 0\)
prsb	spinning reserve balance	\(u_{g,d} p_{g, max} 1_{tl} - z_{u_{g}} - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
actv	startup action	\(u_{g,d} M_{r} - v_{g,d}[:, 1:] = 0\)
actv0	initial startup action	\(u_{g,d}[:, 0] - u_{g}[:, 0] - v_{g,d}[:, 0] = 0\)
actw	shutdown action	\(-u_{g,d} M_{r} - w_{g,d}[:, 1:] = 0\)
actw0	initial shutdown action	\(-u_{g,d}[:, 0] + u_{g}[:, 0] - w_{g,d}[:, 0] = 0\)
zuglb	zug lower bound	\(- z_{u_{g}} + p_g \leq 0\)
zugub	zug upper bound	\(z_{u_{g}} - p_g - M_{zug} (1 - u_{g,d}) \leq 0\)
zugub2	zug upper bound	\(z_{u_{g}} - M_{zug} u_{g,d} \leq 0\)
don	minimum online duration	\(T_{on} v_{g,d} - u_{g,d} \leq 0\)
doff	minimum offline duration	\(T_{off} w_{g,d} - (1 - u_{g,d}) \leq 0\)
pdumax	unserved demand upper bound	\(p_{d,u} - p_{d,s}^{+} c_{trl,d} 1_{tl} \leq 0\)
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
prns	\(p_{r, ns}\)	2D Non-spinning reserve		StaticGen	nonneg
prs	\(p_{r,s}\)	2D Spinning reserve	p.u.	StaticGen	nonneg
ugd	\(u_{g,d}\)	commitment decision		StaticGen.u	boolean
vgd	\(v_{g,d}\)	startup action		StaticGen.u	boolean
wgd	\(w_{g,d}\)	shutdown action		StaticGen.u	boolean
zug	\(z_{ug}\)	Aux var, \(z_{ug} = u_{g,d} * p_g\)		StaticGen	pos
pdu	\(p_{d,u}\)	unserved demand	p.u.	StaticLoad	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	\(c_{trl, e}\)	Reshaped controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Reshaped non-controllability	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
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
dnsrpz	\(d_{nsr, p, z}\)	zonal non-spinning reserve requirement in percentage	NumOpDual
dnsr	\(d_{nsr}\)	zonal non-spinning reserve requirement	NumOpDual
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
Mzug	\(M_{zug}\)	10 times of max of pmax as big M for zug	NumOp
Con	\(T_{on}\)	minimum ON coefficient	MinDur
Coff	\(T_{off}\)	minimum OFF coefficient	MinDur
pdsp	\(p_{d,s}^{+}\)	positive demand	NumOp
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
timeslot	\(t_{s,idx}\)	Time slot for multi-period UC		UCSlot.idx
sd	\(s_{d}\)	area load scaling factor for UC		UCSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
cnsr	\(c_{nsr}\)	cost for non-spinning reserve	$/(p.u.*h)	NSRCost.cnsr
dnsr	\(d_{nsr}\)	non-spinning reserve requirement in percentage	%	NSR.demand
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
csu	\(c_{su}\)	startup cost	$	GCost.csu
csd	\(c_{sd}\)	shutdown cost	$	GCost.csd
cdp	\(c_{d,p}\)	penalty for unserved load	$/(p.u.*h)	DCost.cdp
dctrl	\(c_{trl,d}\)	load controllability		StaticLoad.ctrl
td1	\(t_{d1}\)	minimum ON duration	h	StaticGen.td1
td2	\(t_{d2}\)	minimum OFF duration	h	StaticGen.td2
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 [UCDG]

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

UCES

UC with energy storage ESD1.

Objective

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

Expressions

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

Constraints

Name	Description	Expression
pb	power balance	\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} (p_{d,s}-p_{d,u}) + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\)
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	\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
prnsb	non-spinning reserve balance	\(1-u_{g,d} p_{g, max} 1_{tl} - p_{r, ns} = 0\)
rnsr	non-spinning reserve requirement	\(-S_{g} p_{r, ns} + d_{nsr} \leq 0\)
prsb	spinning reserve balance	\(u_{g,d} p_{g, max} 1_{tl} - z_{u_{g}} - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
actv	startup action	\(u_{g,d} M_{r} - v_{g,d}[:, 1:] = 0\)
actv0	initial startup action	\(u_{g,d}[:, 0] - u_{g}[:, 0] - v_{g,d}[:, 0] = 0\)
actw	shutdown action	\(-u_{g,d} M_{r} - w_{g,d}[:, 1:] = 0\)
actw0	initial shutdown action	\(-u_{g,d}[:, 0] + u_{g}[:, 0] - w_{g,d}[:, 0] = 0\)
zuglb	zug lower bound	\(- z_{u_{g}} + p_g \leq 0\)
zugub	zug upper bound	\(z_{u_{g}} - p_g - M_{zug} (1 - u_{g,d}) \leq 0\)
zugub2	zug upper bound	\(z_{u_{g}} - M_{zug} u_{g,d} \leq 0\)
don	minimum online duration	\(T_{on} v_{g,d} - u_{g,d} \leq 0\)
doff	minimum offline duration	\(T_{off} w_{g,d} - (1 - u_{g,d}) \leq 0\)
pdumax	unserved demand upper bound	\(p_{d,u} - p_{d,s}^{+} c_{trl,d} 1_{tl} \leq 0\)
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
prns	\(p_{r, ns}\)	2D Non-spinning reserve		StaticGen	nonneg
prs	\(p_{r,s}\)	2D Spinning reserve	p.u.	StaticGen	nonneg
ugd	\(u_{g,d}\)	commitment decision		StaticGen.u	boolean
vgd	\(v_{g,d}\)	startup action		StaticGen.u	boolean
wgd	\(w_{g,d}\)	shutdown action		StaticGen.u	boolean
zug	\(z_{ug}\)	Aux var, \(z_{ug} = u_{g,d} * p_g\)		StaticGen	pos
pdu	\(p_{d,u}\)	unserved demand	p.u.	StaticLoad	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	\(c_{trl, e}\)	Reshaped controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Reshaped non-controllability	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
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
dnsrpz	\(d_{nsr, p, z}\)	zonal non-spinning reserve requirement in percentage	NumOpDual
dnsr	\(d_{nsr}\)	zonal non-spinning reserve requirement	NumOpDual
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
Mzug	\(M_{zug}\)	10 times of max of pmax as big M for zug	NumOp
Con	\(T_{on}\)	minimum ON coefficient	MinDur
Coff	\(T_{off}\)	minimum OFF coefficient	MinDur
pdsp	\(p_{d,s}^{+}\)	positive demand	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	\(u_{g}\)	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
timeslot	\(t_{s,idx}\)	Time slot for multi-period UC		UCSlot.idx
sd	\(s_{d}\)	area load scaling factor for UC		UCSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
cnsr	\(c_{nsr}\)	cost for non-spinning reserve	$/(p.u.*h)	NSRCost.cnsr
dnsr	\(d_{nsr}\)	non-spinning reserve requirement in percentage	%	NSR.demand
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
csu	\(c_{su}\)	startup cost	$	GCost.csu
csd	\(c_{sd}\)	shutdown cost	$	GCost.csd
cdp	\(c_{d,p}\)	penalty for unserved load	$/(p.u.*h)	DCost.cdp
dctrl	\(c_{trl,d}\)	load controllability		StaticLoad.ctrl
td1	\(t_{d1}\)	minimum ON duration	h	StaticGen.td1
td2	\(t_{d2}\)	minimum OFF duration	h	StaticGen.td2
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 [UCES]

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

UC2

DC-based unit commitment (UC) using PTDF formulations: The bilinear term in the formulation is linearized with big-M method.

Non-negative var pdu is introduced as unserved load with its penalty cdp.

Constraints include power balance, ramping, spinning reserve, non-spinning reserve, minimum ON/OFF duration. The cost inludes generation cost, startup cost, shutdown cost, spinning reserve cost, non-spinning reserve cost, and unserved load penalty.

Method _initial_guess is used to make initial guess for commitment decision if all generators are online at initial. It is a simple heuristic method, which may not be optimal.

Notes

• The formulations have been adjusted with interval config.t

• The tie-line flow has not been implemented in formulations.

References

1. Huang, Y., Pardalos, P. M., & Zheng, Q. P. (2017). Electrical power unit commitment: deterministic and two-stage stochastic programming models and algorithms. Springer.

2. D. A. Tejada-Arango, S. Lumbreras, P. Sánchez-Martín and A. Ramos, "Which Unit-Commitment Formulation is Best? A Comparison Framework," in IEEE Transactions on Power Systems, vol. 35, no. 4, pp. 2926-2936, July 2020, doi: 10.1109/TPWRS.2019.2962024.

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g)+ \sum(u_{g} c_{0} 1_{tl})+ \sum(c_{su} v_{g,d} + c_{sd} w_{g,d}) + T_{cfg} \sum(c_{sr} p_{r,s} + c_{nsr} p_{r, ns} + c_{d,p} p_{d,u})\)

Expressions

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

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g, axis=0) - \sum(p_{d,s} - p_{d,u}, axis=0) = 0\)
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	\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
prnsb	non-spinning reserve balance	\(1-u_{g,d} p_{g, max} 1_{tl} - p_{r, ns} = 0\)
rnsr	non-spinning reserve requirement	\(-S_{g} p_{r, ns} + d_{nsr} \leq 0\)
prsb	spinning reserve balance	\(u_{g,d} p_{g, max} 1_{tl} - z_{u_{g}} - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
actv	startup action	\(u_{g,d} M_{r} - v_{g,d}[:, 1:] = 0\)
actv0	initial startup action	\(u_{g,d}[:, 0] - u_{g}[:, 0] - v_{g,d}[:, 0] = 0\)
actw	shutdown action	\(-u_{g,d} M_{r} - w_{g,d}[:, 1:] = 0\)
actw0	initial shutdown action	\(-u_{g,d}[:, 0] + u_{g}[:, 0] - w_{g,d}[:, 0] = 0\)
zuglb	zug lower bound	\(- z_{u_{g}} + p_g \leq 0\)
zugub	zug upper bound	\(z_{u_{g}} - p_g - M_{zug} (1 - u_{g,d}) \leq 0\)
zugub2	zug upper bound	\(z_{u_{g}} - M_{zug} u_{g,d} \leq 0\)
don	minimum online duration	\(T_{on} v_{g,d} - u_{g,d} \leq 0\)
doff	minimum offline duration	\(T_{off} w_{g,d} - (1 - u_{g,d}) \leq 0\)
pdumax	unserved demand upper bound	\(p_{d,u} - p_{d,s}^{+} c_{trl,d} 1_{tl} \leq 0\)

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
prns	\(p_{r, ns}\)	2D Non-spinning reserve		StaticGen	nonneg
prs	\(p_{r,s}\)	2D Spinning reserve	p.u.	StaticGen	nonneg
ugd	\(u_{g,d}\)	commitment decision		StaticGen.u	boolean
vgd	\(v_{g,d}\)	startup action		StaticGen.u	boolean
wgd	\(w_{g,d}\)	shutdown action		StaticGen.u	boolean
zug	\(z_{ug}\)	Aux var, \(z_{ug} = u_{g,d} * p_g\)		StaticGen	pos
pdu	\(p_{d,u}\)	unserved demand	p.u.	StaticLoad	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	\(c_{trl, e}\)	Reshaped controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Reshaped non-controllability	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
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
dnsrpz	\(d_{nsr, p, z}\)	zonal non-spinning reserve requirement in percentage	NumOpDual
dnsr	\(d_{nsr}\)	zonal non-spinning reserve requirement	NumOpDual
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
Mzug	\(M_{zug}\)	10 times of max of pmax as big M for zug	NumOp
Con	\(T_{on}\)	minimum ON coefficient	MinDur
Coff	\(T_{off}\)	minimum OFF coefficient	MinDur
pdsp	\(p_{d,s}^{+}\)	positive demand	NumOp
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
timeslot	\(t_{s,idx}\)	Time slot for multi-period UC		UCSlot.idx
sd	\(s_{d}\)	area load scaling factor for UC		UCSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
cnsr	\(c_{nsr}\)	cost for non-spinning reserve	$/(p.u.*h)	NSRCost.cnsr
dnsr	\(d_{nsr}\)	non-spinning reserve requirement in percentage	%	NSR.demand
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
csu	\(c_{su}\)	startup cost	$	GCost.csu
csd	\(c_{sd}\)	shutdown cost	$	GCost.csd
cdp	\(c_{d,p}\)	penalty for unserved load	$/(p.u.*h)	DCost.cdp
dctrl	\(c_{trl,d}\)	load controllability		StaticLoad.ctrl
td1	\(t_{d1}\)	minimum ON duration	h	StaticGen.td1
td2	\(t_{d2}\)	minimum OFF duration	h	StaticGen.td2
PTDF	\(P_{TDF}\)	PTDF		MatProcessor.PTDF

Config Fields in [UC2]

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

UC2DG

UC with distributed generation DG, using PTDF formulations.

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

Objective

Unit	Expression
$	\(min. T_{cfg}^{2} \sum(c_{2} p_g^{2})+ T_{cfg} \sum(c_{1} p_g)+ \sum(u_{g} c_{0} 1_{tl})+ \sum(c_{su} v_{g,d} + c_{sd} w_{g,d}) + T_{cfg} \sum(c_{sr} p_{r,s} + c_{nsr} p_{r, ns} + c_{d,p} p_{d,u})\)

Expressions

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

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g, axis=0) - \sum(p_{d,s} - p_{d,u}, axis=0) = 0\)
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	\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
prnsb	non-spinning reserve balance	\(1-u_{g,d} p_{g, max} 1_{tl} - p_{r, ns} = 0\)
rnsr	non-spinning reserve requirement	\(-S_{g} p_{r, ns} + d_{nsr} \leq 0\)
prsb	spinning reserve balance	\(u_{g,d} p_{g, max} 1_{tl} - z_{u_{g}} - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
actv	startup action	\(u_{g,d} M_{r} - v_{g,d}[:, 1:] = 0\)
actv0	initial startup action	\(u_{g,d}[:, 0] - u_{g}[:, 0] - v_{g,d}[:, 0] = 0\)
actw	shutdown action	\(-u_{g,d} M_{r} - w_{g,d}[:, 1:] = 0\)
actw0	initial shutdown action	\(-u_{g,d}[:, 0] + u_{g}[:, 0] - w_{g,d}[:, 0] = 0\)
zuglb	zug lower bound	\(- z_{u_{g}} + p_g \leq 0\)
zugub	zug upper bound	\(z_{u_{g}} - p_g - M_{zug} (1 - u_{g,d}) \leq 0\)
zugub2	zug upper bound	\(z_{u_{g}} - M_{zug} u_{g,d} \leq 0\)
don	minimum online duration	\(T_{on} v_{g,d} - u_{g,d} \leq 0\)
doff	minimum offline duration	\(T_{off} w_{g,d} - (1 - u_{g,d}) \leq 0\)
pdumax	unserved demand upper bound	\(p_{d,u} - p_{d,s}^{+} c_{trl,d} 1_{tl} \leq 0\)
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
prns	\(p_{r, ns}\)	2D Non-spinning reserve		StaticGen	nonneg
prs	\(p_{r,s}\)	2D Spinning reserve	p.u.	StaticGen	nonneg
ugd	\(u_{g,d}\)	commitment decision		StaticGen.u	boolean
vgd	\(v_{g,d}\)	startup action		StaticGen.u	boolean
wgd	\(w_{g,d}\)	shutdown action		StaticGen.u	boolean
zug	\(z_{ug}\)	Aux var, \(z_{ug} = u_{g,d} * p_g\)		StaticGen	pos
pdu	\(p_{d,u}\)	unserved demand	p.u.	StaticLoad	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	\(c_{trl, e}\)	Reshaped controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Reshaped non-controllability	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
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
dnsrpz	\(d_{nsr, p, z}\)	zonal non-spinning reserve requirement in percentage	NumOpDual
dnsr	\(d_{nsr}\)	zonal non-spinning reserve requirement	NumOpDual
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
Mzug	\(M_{zug}\)	10 times of max of pmax as big M for zug	NumOp
Con	\(T_{on}\)	minimum ON coefficient	MinDur
Coff	\(T_{off}\)	minimum OFF coefficient	MinDur
pdsp	\(p_{d,s}^{+}\)	positive demand	NumOp
PTDFt	\(P_{TDF}^T\)	PTDF transpose	NumOp
idg	\(I_{DG}\)	Index DG power from pg	VarSelect

Parameters

Name	Symbol	Description	Unit	Source
ug	\(u_{g}\)	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
timeslot	\(t_{s,idx}\)	Time slot for multi-period UC		UCSlot.idx
sd	\(s_{d}\)	area load scaling factor for UC		UCSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
cnsr	\(c_{nsr}\)	cost for non-spinning reserve	$/(p.u.*h)	NSRCost.cnsr
dnsr	\(d_{nsr}\)	non-spinning reserve requirement in percentage	%	NSR.demand
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
csu	\(c_{su}\)	startup cost	$	GCost.csu
csd	\(c_{sd}\)	shutdown cost	$	GCost.csd
cdp	\(c_{d,p}\)	penalty for unserved load	$/(p.u.*h)	DCost.cdp
dctrl	\(c_{trl,d}\)	load controllability		StaticLoad.ctrl
td1	\(t_{d1}\)	minimum ON duration	h	StaticGen.td1
td2	\(t_{d2}\)	minimum OFF duration	h	StaticGen.td2
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 [UC2DG]

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

UC2ES

UC with energy storage ESD1, using PTDF formulations.

Objective

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

Expressions

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

Constraints

Name	Description	Expression
pb	power balance	\(\sum(p_g, axis=0) - \sum(p_{d,s} - p_{d,u}, axis=0) = 0\)
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	\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
plfub	line flow upper bound	\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} 1_{tl} \leq 0\)
alflb	line angle difference lower bound	\(-C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
alfub	line angle difference upper bound	\(C_{ft}^T \theta_{bus} - \theta_{bus, max} 1_{tl} \leq 0\)
prnsb	non-spinning reserve balance	\(1-u_{g,d} p_{g, max} 1_{tl} - p_{r, ns} = 0\)
rnsr	non-spinning reserve requirement	\(-S_{g} p_{r, ns} + d_{nsr} \leq 0\)
prsb	spinning reserve balance	\(u_{g,d} p_{g, max} 1_{tl} - z_{u_{g}} - p_{r,s} = 0\)
rsr	spinning reserve requirement	\(-S_{g} p_{r,s} + d_{s,r,z} \leq 0\)
actv	startup action	\(u_{g,d} M_{r} - v_{g,d}[:, 1:] = 0\)
actv0	initial startup action	\(u_{g,d}[:, 0] - u_{g}[:, 0] - v_{g,d}[:, 0] = 0\)
actw	shutdown action	\(-u_{g,d} M_{r} - w_{g,d}[:, 1:] = 0\)
actw0	initial shutdown action	\(-u_{g,d}[:, 0] + u_{g}[:, 0] - w_{g,d}[:, 0] = 0\)
zuglb	zug lower bound	\(- z_{u_{g}} + p_g \leq 0\)
zugub	zug upper bound	\(z_{u_{g}} - p_g - M_{zug} (1 - u_{g,d}) \leq 0\)
zugub2	zug upper bound	\(z_{u_{g}} - M_{zug} u_{g,d} \leq 0\)
don	minimum online duration	\(T_{on} v_{g,d} - u_{g,d} \leq 0\)
doff	minimum offline duration	\(T_{off} w_{g,d} - (1 - u_{g,d}) \leq 0\)
pdumax	unserved demand upper bound	\(p_{d,u} - p_{d,s}^{+} c_{trl,d} 1_{tl} \leq 0\)
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
prns	\(p_{r, ns}\)	2D Non-spinning reserve		StaticGen	nonneg
prs	\(p_{r,s}\)	2D Spinning reserve	p.u.	StaticGen	nonneg
ugd	\(u_{g,d}\)	commitment decision		StaticGen.u	boolean
vgd	\(v_{g,d}\)	startup action		StaticGen.u	boolean
wgd	\(w_{g,d}\)	shutdown action		StaticGen.u	boolean
zug	\(z_{ug}\)	Aux var, \(z_{ug} = u_{g,d} * p_g\)		StaticGen	pos
pdu	\(p_{d,u}\)	unserved demand	p.u.	StaticLoad	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	\(c_{trl, e}\)	Reshaped controllability	NumOpDual
nctrl	\(c_{trl,n}\)	Effective Gen uncontrollability	NumOp
nctrle	\(c_{trl,n,e}\)	Reshaped non-controllability	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
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
dnsrpz	\(d_{nsr, p, z}\)	zonal non-spinning reserve requirement in percentage	NumOpDual
dnsr	\(d_{nsr}\)	zonal non-spinning reserve requirement	NumOpDual
dsrpz	\(d_{s,r, p, z}\)	zonal spinning reserve requirement in percentage	NumOpDual
dsr	\(d_{s,r,z}\)	zonal spinning reserve requirement	NumOpDual
Mzug	\(M_{zug}\)	10 times of max of pmax as big M for zug	NumOp
Con	\(T_{on}\)	minimum ON coefficient	MinDur
Coff	\(T_{off}\)	minimum OFF coefficient	MinDur
pdsp	\(p_{d,s}^{+}\)	positive demand	NumOp
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	\(u_{g}\)	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
timeslot	\(t_{s,idx}\)	Time slot for multi-period UC		UCSlot.idx
sd	\(s_{d}\)	area load scaling factor for UC		UCSlotLoad.sd
R30	\(R_{30}\)	30-min ramp rate	p.u./h	StaticGen.R30
cnsr	\(c_{nsr}\)	cost for non-spinning reserve	$/(p.u.*h)	NSRCost.cnsr
dnsr	\(d_{nsr}\)	non-spinning reserve requirement in percentage	%	NSR.demand
dsr	\(d_{sr}\)	spinning reserve requirement in percentage	%	SR.demand
csr	\(c_{sr}\)	cost for spinning reserve	$/(p.u.*h)	SRCost.csr
csu	\(c_{su}\)	startup cost	$	GCost.csu
csd	\(c_{sd}\)	shutdown cost	$	GCost.csd
cdp	\(c_{d,p}\)	penalty for unserved load	$/(p.u.*h)	DCost.cdp
dctrl	\(c_{trl,d}\)	load controllability		StaticLoad.ctrl
td1	\(t_{d1}\)	minimum ON duration	h	StaticGen.td1
td2	\(t_{d2}\)	minimum OFF duration	h	StaticGen.td2
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 [UC2ES]

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