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, ED, EDDG, EDES, RTED, RTEDDG, RTEDES, RTEDVIS
DCOPF#
DC optimal power flow (DCOPF).
Line flow variable plf is calculated as Bf@aBus + Pfinj
after solving the problem in _post_solve() .
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total cost |
$ |
\(min. \sum(c_{2} power(p_g, 2))+ \sum(c_{1} p_g)+ \sum(u_{g} c_{0})\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} - c_{trl, e} p_{g, max} <= 0\) |
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 |
\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
Gen active power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
Bus voltage angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
Line flow |
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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
ED#
DC-based multi-period economic dispatch (ED).
Dispath interval config.t (\(T_{cfg}\)) is introduced,
1 [Hour] by default.
ED extends DCOPF as follows:
Vars
pg,pru,prdare extended to 2D2D Vars
rguandrgdare introducedParam
ugis sourced fromEDTSlot.ugas commitment decisions
Notes#
Formulations has been adjusted with interval
config.tThe tie-line flow is not implemented in this model.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} (T_{cfg} p_g)^{2} + c_{1} (T_{cfg} p_g))+ \sum(u_{g} c_{0} 1_{tl}) + \sum(c_{sr} (T_{cfg} p_{r,s}))\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} 1_{tl} - c_{trl, e} 1_{tl} p_{g, max} <= 0\) |
pb |
power balance |
\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} p_{d,s} + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\) |
plflb |
line flow lower bound |
\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - R_{ATEA} 1_{tl} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - R_{ATEA} 1_{tl} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} 1_{tl} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} 1_{tl} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - p_{g, max} 1_{tl} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + p_{g, min} 1_{tl} <= 0\) |
rgu |
Gen ramping up |
\(p_g M_{r} - T_{cfg} R_{30,R} <= 0\) |
rgd |
Gen ramping down |
\(-p_g M_{r} - T_{cfg} R_{30,R} <= 0\) |
prsb |
spinning reserve balance |
\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\) |
rsr |
spinning reserve requirement |
\(-S_{g} p_{r,s} + d_{s,r,z} <= 0\) |
rgu0 |
Initial gen ramping up |
\(p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} <= 0\) |
rgd0 |
Initial gen ramping down |
\(- p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} <= 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
2D Gen power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
2D Bus angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
2D Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
2D RegUp power |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
2D RegDn power |
p.u. |
nonneg |
|
prs |
\(p_{r,s}\) |
spinning reserve |
p.u. |
nonneg |
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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 |
ugt |
\(u_{g}\) |
input ug transpose |
NumOp |
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 |
\(u_{g}\) |
unit commitment decisions |
EDTSlot.ug |
|
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |
sd |
\(s_{d}\) |
zonal load factor for ED |
EDTSlot.sd |
|
timeslot |
\(t_{s,idx}\) |
Time slot for multi-period ED |
EDTSlot.idx |
|
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 |
EDDG#
ED with distributed generation DG.
Note that EDDG only inlcudes DG output power. If ESD1 is included, EDES should be used instead, otherwise there is no SOC.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} (T_{cfg} p_g)^{2} + c_{1} (T_{cfg} p_g))+ \sum(u_{g} c_{0} 1_{tl}) + \sum(c_{sr} (T_{cfg} p_{r,s}))\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} 1_{tl} - c_{trl, e} 1_{tl} p_{g, max} <= 0\) |
pb |
power balance |
\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} p_{d,s} + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\) |
plflb |
line flow lower bound |
\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - R_{ATEA} 1_{tl} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - R_{ATEA} 1_{tl} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} 1_{tl} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} 1_{tl} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - p_{g, max} 1_{tl} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + p_{g, min} 1_{tl} <= 0\) |
rgu |
Gen ramping up |
\(p_g M_{r} - T_{cfg} R_{30,R} <= 0\) |
rgd |
Gen ramping down |
\(-p_g M_{r} - T_{cfg} R_{30,R} <= 0\) |
prsb |
spinning reserve balance |
\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\) |
rsr |
spinning reserve requirement |
\(-S_{g} p_{r,s} + d_{s,r,z} <= 0\) |
rgu0 |
Initial gen ramping up |
\(p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} <= 0\) |
rgd0 |
Initial gen ramping down |
\(- p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} <= 0\) |
cdgb |
Select DG power from pg |
\(C_{DG} p_g - p_{g,DG} = 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
2D Gen power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
2D Bus angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
2D Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
2D RegUp power |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
2D RegDn power |
p.u. |
nonneg |
|
prs |
\(p_{r,s}\) |
spinning reserve |
p.u. |
nonneg |
|
pgdg |
\(p_{g,DG}\) |
DG output 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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 |
ugt |
\(u_{g}\) |
input ug transpose |
NumOp |
cd |
\(C_{DG}\) |
Select DG power from pg |
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 |
\(u_{g}\) |
unit commitment decisions |
EDTSlot.ug |
|
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |
sd |
\(s_{d}\) |
zonal load factor for ED |
EDTSlot.sd |
|
timeslot |
\(t_{s,idx}\) |
Time slot for multi-period ED |
EDTSlot.idx |
|
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 |
EDES#
ED with energy storage ESD1. The bilinear term in the formulation is linearized with big-M method.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} (T_{cfg} p_g)^{2} + c_{1} (T_{cfg} p_g))+ \sum(u_{g} c_{0} 1_{tl}) + \sum(c_{sr} (T_{cfg} p_{r,s}))\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} 1_{tl} + c_{trl, e} 1_{tl} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} 1_{tl} - c_{trl, e} 1_{tl} p_{g, max} <= 0\) |
pb |
power balance |
\(B_{bus} \theta_{bus} + P_{bus}^{inj} 1_{tl} + C_{l} p_{d,s} + C_{sh} g_{sh} 1_{tl} - C_{g} p_g = 0\) |
plflb |
line flow lower bound |
\(-B_{f} \theta_{bus} - P_{f}^{inj} 1_{tl} - R_{ATEA} 1_{tl} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} 1_{tl} - R_{ATEA} 1_{tl} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} 1_{tl} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} 1_{tl} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - p_{g, max} 1_{tl} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + p_{g, min} 1_{tl} <= 0\) |
rgu |
Gen ramping up |
\(p_g M_{r} - T_{cfg} R_{30,R} <= 0\) |
rgd |
Gen ramping down |
\(-p_g M_{r} - T_{cfg} R_{30,R} <= 0\) |
prsb |
spinning reserve balance |
\(u_{g} p_{g, max} 1_{tl} - p_g - p_{r,s} = 0\) |
rsr |
spinning reserve requirement |
\(-S_{g} p_{r,s} + d_{s,r,z} <= 0\) |
rgu0 |
Initial gen ramping up |
\(p_g[:, 0] - p_{g, 0}[:, 0] - R_{30} <= 0\) |
rgd0 |
Initial gen ramping down |
\(- p_g[:, 0] + p_{g, 0}[:, 0] - R_{30} <= 0\) |
cdgb |
Select DG power from pg |
\(C_{DG} p_g - p_{g,DG} = 0\) |
SOClb |
SOC lower bound |
\(-SOC + SOC_{min} <= 0\) |
SOCub |
SOC upper bound |
\(SOC - SOC_{max} <= 0\) |
cdb |
Charging decision bound |
\(u_{c,ESD} + u_{d,ESD} - 1 = 0\) |
cesb |
Select ESD1 power from pg |
\(C_{ESD} p_g + z_{c,ESD} - z_{d,ESD} = 0\) |
zce1 |
zce bound 1 |
\(-z_{c,ESD} + p_{c,ESD} <= 0\) |
zce2 |
zce bound 2 |
\(z_{c,ESD} - p_{c,ESD} - M_{big} (1-u_{c,ESD}) <= 0\) |
zce3 |
zce bound 3 |
\(z_{c,ESD} - M_{big} u_{c,ESD} <= 0\) |
zde1 |
zde bound 1 |
\(-z_{d,ESD} + p_{d,ESD} <= 0\) |
zde2 |
zde bound 2 |
\(z_{d,ESD} - p_{d,ESD} - M_{big} (1-u_{d,ESD}) <= 0\) |
zde3 |
zde bound 3 |
\(z_{d,ESD} - M_{big} u_{d,ESD} <= 0\) |
SOCb |
ESD1 SOC balance |
\(E_{n,R} SOC M_{r,ES} - T_{cfg} \eta_{c,R} z_{c,ESD}[:, 1:] + T_{cfg} R_{\eta_d,R} z_{d,ESD}[:, 1:] = 0\) |
SOCb0 |
ESD1 SOC initial balance |
\(E_n SOC[:, 0] - SOC_{init} - T_{cfg} \eta_c z_{c,ESD}[:, 0] + T_{cfg} \frac{1}{\eta_d} z_{d,ESD}[:, 0] = 0\) |
SOCr |
SOC requirement |
\(SOC[:, -1] - SOC_{init} = 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
2D Gen power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
2D Bus angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
2D Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
2D RegUp power |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
2D RegDn power |
p.u. |
nonneg |
|
prs |
\(p_{r,s}\) |
spinning reserve |
p.u. |
nonneg |
|
pgdg |
\(p_{g,DG}\) |
DG output power |
p.u. |
||
SOC |
\(SOC\) |
ESD1 State of Charge |
% |
pos |
|
pce |
\(p_{c,ESD}\) |
ESD1 charging power |
p.u. |
nonneg |
|
pde |
\(p_{d,ESD}\) |
ESD1 discharging power |
p.u. |
nonneg |
|
uce |
\(u_{c,ESD}\) |
ESD1 charging decision |
boolean |
||
ude |
\(u_{d,ESD}\) |
ESD1 discharging decision |
boolean |
||
zce |
\(z_{c,ESD}\) |
Aux var for charging, \(z_{c,ESD}=u_{c,ESD}*p_{c,ESD}\) |
nonneg |
||
zde |
\(z_{d,ESD}\) |
Aux var for discharging, \(z_{d,ESD}=u_{d,ESD}*p_{d,ESD}\) |
nonneg |
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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 |
ugt |
\(u_{g}\) |
input ug transpose |
NumOp |
cd |
\(C_{DG}\) |
Select DG power from pg |
VarSelect |
REtaD |
\(\frac{1}{\eta_d}\) |
NumOp |
|
Mb |
\(M_{big}\) |
10 times of max of pmax as big M |
NumOp |
ce |
\(C_{ESD}\) |
Select zue 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 |
|---|---|---|---|---|
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 |
\(u_{g}\) |
unit commitment decisions |
EDTSlot.ug |
|
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |
sd |
\(s_{d}\) |
zonal load factor for ED |
EDTSlot.sd |
|
timeslot |
\(t_{s,idx}\) |
Time slot for multi-period ED |
EDTSlot.idx |
|
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 |
|
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 |
EtaC |
\(\eta_c\) |
Efficiency during charging |
% |
ESD1.EtaC |
EtaD |
\(\eta_d\) |
Efficiency during discharging |
% |
ESD1.EtaD |
genesd |
\(g_{ESD}\) |
gen of ESD1 |
ESD1.gen |
|
gammapesd |
\(\gamma_{p,ESD}\) |
Ratio of ESD1.pge w.r.t to that of static generator |
ESD1.gammap |
RTED#
DC-based real-time economic dispatch (RTED). RTED extends DCOPF with:
Mapping dicts to interface with ANDES
Function
dc2acto do the AC conversionVars for SFR reserve:
pruandprdParam for linear SFR cost:
cruandcrdParam for SFR requirement:
duandddParam for ramping: start point
pg0and ramping limitR10Param
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 has been adjusted with interval
config.t, 5/60 [Hour] by default.The tie-line flow has not been implemented in formulations.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} power(T_{cfg} p_g, 2))+ \sum(c_{1} (T_{cfg} p_g))+ u_{g} c_{0}+ \sum(c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} - c_{trl, e} p_{g, max} <= 0\) |
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 |
\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - u_{g} p_{g, max} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + u_{g} p_{g, min} <= 0\) |
rgu |
Gen ramping up |
\(u_{g} p_g-p_{g, 0}-R_{10} <= 0\) |
rgd |
Gen ramping down |
\(u_{g} -p_g+p_{g, 0}-R_{10} <= 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
Gen active power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
Bus voltage angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
RegUp reserve |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
RegDn reserve |
p.u. |
nonneg |
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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 |
|---|---|---|---|---|
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |
RTEDDG#
RTED with distributed generator DG.
Note that RTEDDG only inlcudes DG output power. If ESD1 is included, RTEDES should be used instead, otherwise there is no SOC.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} power(T_{cfg} p_g, 2))+ \sum(c_{1} (T_{cfg} p_g))+ u_{g} c_{0}+ \sum(c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} - c_{trl, e} p_{g, max} <= 0\) |
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 |
\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - u_{g} p_{g, max} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + u_{g} p_{g, min} <= 0\) |
rgu |
Gen ramping up |
\(u_{g} p_g-p_{g, 0}-R_{10} <= 0\) |
rgd |
Gen ramping down |
\(u_{g} -p_g+p_{g, 0}-R_{10} <= 0\) |
cdgb |
Select DG power from pg |
\(C_{DG} p_g - p_{g,DG} = 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
Gen active power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
Bus voltage angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
RegUp reserve |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
RegDn reserve |
p.u. |
nonneg |
|
pgdg |
\(p_{g,DG}\) |
DG output 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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 |
cd |
\(C_{DG}\) |
Select DG power from pg |
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |
RTEDES#
RTED with energy storage ESD1. The bilinear term in the formulation is linearized with big-M method.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} power(T_{cfg} p_g, 2))+ \sum(c_{1} (T_{cfg} p_g))+ u_{g} c_{0}+ \sum(c_{r,u} p_{r,u} + c_{r,d} p_{r,d})\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} - c_{trl, e} p_{g, max} <= 0\) |
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 |
\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - u_{g} p_{g, max} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + u_{g} p_{g, min} <= 0\) |
rgu |
Gen ramping up |
\(u_{g} p_g-p_{g, 0}-R_{10} <= 0\) |
rgd |
Gen ramping down |
\(u_{g} -p_g+p_{g, 0}-R_{10} <= 0\) |
cdgb |
Select DG power from pg |
\(C_{DG} p_g - p_{g,DG} = 0\) |
SOClb |
SOC lower bound |
\(-SOC + SOC_{min} <= 0\) |
SOCub |
SOC upper bound |
\(SOC - SOC_{max} <= 0\) |
cdb |
Charging decision bound |
\(u_{c,ESD} + u_{d,ESD} - 1 = 0\) |
cesb |
Select ESD1 power from pg |
\(C_{ESD} p_g + z_{c,ESD} - z_{d,ESD} = 0\) |
zce1 |
zce bound 1 |
\(-z_{c,ESD} + p_{c,ESD} <= 0\) |
zce2 |
zce bound 2 |
\(z_{c,ESD} - p_{c,ESD} - M_{big} (1-u_{c,ESD}) <= 0\) |
zce3 |
zce bound 3 |
\(z_{c,ESD} - M_{big} u_{c,ESD} <= 0\) |
zde1 |
zde bound 1 |
\(-z_{d,ESD} + p_{d,ESD} <= 0\) |
zde2 |
zde bound 2 |
\(z_{d,ESD} - p_{d,ESD} - M_{big} (1-u_{d,ESD}) <= 0\) |
zde3 |
zde bound 3 |
\(z_{d,ESD} - M_{big} u_{d,ESD} <= 0\) |
SOCb |
ESD1 SOC balance |
\(E_n (SOC - SOC_{init}) - T_{cfg} \eta_c z_{c,ESD}+ T_{cfg} \frac{1}{\eta_d} z_{d,ESD} = 0\) |
Vars#
Name |
Symbol |
Description |
Unit |
Source |
Properties |
|---|---|---|---|---|---|
pg |
\(p_g\) |
Gen active power |
p.u. |
StaticGen.p |
|
aBus |
\(\theta_{bus}\) |
Bus voltage angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
RegUp reserve |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
RegDn reserve |
p.u. |
nonneg |
|
pgdg |
\(p_{g,DG}\) |
DG output power |
p.u. |
||
SOC |
\(SOC\) |
ESD1 State of Charge |
% |
pos |
|
pce |
\(p_{c,ESD}\) |
ESD1 charging power |
p.u. |
nonneg |
|
pde |
\(p_{d,ESD}\) |
ESD1 discharging power |
p.u. |
nonneg |
|
uce |
\(u_{c,ESD}\) |
ESD1 charging decision |
boolean |
||
ude |
\(u_{d,ESD}\) |
ESD1 discharging decision |
boolean |
||
zce |
\(z_{c,ESD}\) |
Aux var for charging, \(z_{c,ESD}=u_{c,ESD}*p_{c,ESD}\) |
nonneg |
||
zde |
\(z_{d,ESD}\) |
Aux var for discharging, \(z_{d,ESD}=u_{d,ESD}*p_{d,ESD}\) |
nonneg |
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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 |
cd |
\(C_{DG}\) |
Select DG power from pg |
VarSelect |
REtaD |
\(\frac{1}{\eta_d}\) |
NumOp |
|
Mb |
\(M_{big}\) |
10 times of max of pmax as big M |
NumOp |
ce |
\(C_{ESD}\) |
Select zue from pg |
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |
EtaC |
\(\eta_c\) |
Efficiency during charging |
% |
ESD1.EtaC |
EtaD |
\(\eta_d\) |
Efficiency during discharging |
% |
ESD1.EtaD |
genesd |
\(g_{ESD}\) |
gen of ESD1 |
ESD1.gen |
|
gammapesd |
\(\gamma_{p,ESD}\) |
Ratio of ESD1.pge w.r.t to that of static generator |
ESD1.gammap |
RTEDVIS#
RTED with virtual inertia scheduling.
Reference:
[1] B. She, F. Li, H. Cui, J. Wang, Q. Zhang and R. Bo, "Virtual Inertia Scheduling (VIS) for Real-time Economic Dispatch of IBRs-penetrated Power Systems," in IEEE Transactions on Sustainable Energy, doi: 10.1109/TSTE.2023.3319307.
Objective#
Name |
Description |
Unit |
Expression |
|---|---|---|---|
obj |
total generation and reserve cost |
$ |
\(min. \sum(c_{2} power(p_g, 2))+ \sum(c_{1} (T_{cfg} p_g))+ u_{g} c_{0} + \sum(c_{r,u} (T_{cfg} p_{r,u}) + c_{r,d} (T_{cfg} p_{r,d})) + \sum(c_{m} M + c_{d} D)\) |
Constraints#
Name |
Description |
Expression |
|---|---|---|
pglb |
pg min |
\(-p_g + c_{trl,n,e} p_{g, 0} + c_{trl, e} p_{g, min} <= 0\) |
pgub |
pg max |
\(p_g - c_{trl,n,e} p_{g, 0} - c_{trl, e} p_{g, max} <= 0\) |
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 |
\(-B_{f} \theta_{bus} - P_{f}^{inj} - R_{ATEA} <= 0\) |
plfub |
line flow upper bound |
\(B_{f} \theta_{bus} + P_{f}^{inj} - R_{ATEA} <= 0\) |
alflb |
line angle difference lower bound |
\(-C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
alfub |
line angle difference upper bound |
\(C_{ft}^T \theta_{bus} - \theta_{max} <= 0\) |
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 |
\(u_{g} p_g + p_{r,u} - u_{g} p_{g, max} <= 0\) |
rrd |
RegDn reserve source |
\(u_{g} -p_g + p_{r,d} + u_{g} p_{g, min} <= 0\) |
rgu |
Gen ramping up |
\(u_{g} p_g-p_{g, 0}-R_{10} <= 0\) |
rgd |
Gen ramping down |
\(u_{g} -p_g+p_{g, 0}-R_{10} <= 0\) |
Mub |
M upper bound |
\(M - M_{max} <= 0\) |
Dub |
D upper bound |
\(D - D_{max} <= 0\) |
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 |
|
aBus |
\(\theta_{bus}\) |
Bus voltage angle |
rad |
Bus.a |
|
plf |
\(p_{lf}\) |
Line flow |
p.u. |
||
pru |
\(p_{r,u}\) |
RegUp reserve |
p.u. |
nonneg |
|
prd |
\(p_{r,d}\) |
RegDn reserve |
p.u. |
nonneg |
|
M |
\(M\) |
Emulated startup time constant (M=2H) |
s |
nonneg |
|
D |
\(D\) |
Emulated damping coefficient |
p.u. |
nonneg |
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 |
amax |
\(\theta_{max}\) |
max line angle difference |
NumOp |
gs |
\(S_{g}\) |
Sum Gen vars vector in shape of zone |
ZonalSum |
ds |
\(S_{d}\) |
Sum pd vector in shape of zone |
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 zone |
ZonalSum |
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 |
pd |
\(p_{d}\) |
active demand |
p.u. |
StaticLoad.p0 |
rate_a |
\(R_{ATEA}\) |
long-term flow limit |
MVA |
Line.rate_a |
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 |
|
zg |
\(z_{one,g}\) |
Gen zone |
StaticGen.zone |
|
zd |
\(z_{one,d}\) |
Load zone |
StaticLoad.zone |
|
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 |