DED#

Type for Distributional economic dispatch.

Available routines: DOPF, DOPFVIS

DOPF#

Linearzied distribution OPF, where power loss are ignored.

UNDER DEVELOPMENT!

Reference:

[1] L. Bai, J. Wang, C. Wang, C. Chen, and F. Li, “Distribution Locational Marginal Pricing (DLMP) for Congestion Management and Voltage Support,” IEEE Trans. Power Syst., vol. 33, no. 4, pp. 4061–4073, Jul. 2018, doi: 10.1109/TPWRS.2017.2767632.

Objective#

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

Constraints#

Name	Description	Expression
pglb	pg min	$-p_g + c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min} \leq 0$
pgub	pg max	$p_g - c_{trl,n,e} p_{g, 0} - c_{trl, e} p_{g, max} \leq 0$
sbus	align slack bus angle	$c_{sb} \theta_{bus} = 0$
pb	power balance	$B_{bus} \theta_{bus} + P_{bus}^{inj} + C_{l} p_{d} + C_{sh} g_{sh} - C_{g} p_g = 0$
plflb	line flow lower bound	$-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} \leq 0$
plfub	line flow upper bound	$B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} \leq 0$
alflb	line angle difference lower bound	$-C_{ft}^T \theta_{bus} + \theta_{bus, min} \leq 0$
alfub	line angle difference upper bound	$C_{ft}^T \theta_{bus} - \theta_{bus, max} \leq 0$
qglb	qg min	$-q_{g} + u_{g} q_{min} \leq 0$
qgub	qg max	$q_{g} - u_{g} q_{max} \leq 0$
vu	Voltage upper limit	$v^{2} - v_{max}^{2} \leq 0$
vl	Voltage lower limit	$-v^{2} + v_{min}^{2} \leq 0$
lvd	line voltage drop	$C_{ft}^T v^{2} - (r p_{lf} + x q_{lf}) = 0$
qb	reactive power balance	$\sum(q_{d}) - \sum(q_{g}) = 0$

Expressions#

Name	Variable	Description	Expression
plfc	plf	plf calculation	$B_{f} \theta_{bus} + P_{f}^{inj}$
pic	pi	dual of Constraint pb	$\phi[pb]$

Vars#

Name	Symbol	Description	Unit	Source
pg	$p_g$	Gen active power	p.u.	StaticGen.p
aBus	$\theta_{bus}$	Bus voltage angle	rad	Bus.a
pi	$\pi$	nodal price	$/p.u.
plf	$p_{lf}$	Line flow	p.u.
qg	$q_{g}$	Gen reactive power	p.u.	StaticGen.q
v	$v$	Bus voltage	p.u.	Bus.v
vsq	$v^{2}$	square of Bus voltage	p.u.
qlf	$q_{lf}$	line reactive power	p.u.

Services#

Name	Symbol	Description	Type
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
csb	$c_{sb}$	select slack bus	VarSelect

Parameters#

Name	Symbol	Description	Unit	Source
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
ug	$u_{g}$	Gen connection status		StaticGen.u
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
p0	$p_{g, 0}$	Gen initial active power	p.u.	StaticGen.pg0
buss	$B_{us,s}$	Bus slack		Slack.bus
pd	$p_{d}$	active demand	p.u.	StaticLoad.p0
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
gsh	$g_{sh}$	shunt conductance		Shunt.g
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
qmax	$q_{max}$	generator maximum reactive power	p.u.	StaticGen.qmax
qmin	$q_{min}$	generator minimum reactive power	p.u.	StaticGen.qmin
qd	$q_{d}$	reactive demand	p.u.	StaticLoad.q0
vmax	$v_{max}$	Bus voltage upper limit	p.u.	Bus.vmax
vmin	$v_{min}$	Bus voltage lower limit	p.u.	Bus.vmin
r	$r$	line resistance	p.u.	Line.r
x	$x$	line reactance	p.u.	Line.x

DOPFVIS#

Linearzied distribution OPF with variables for virtual inertia and damping from from REGCV1, where power loss are ignored.