security-constrained opf and risk-based opf
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Security-Constrained OPFand
Risk-Based OPF
2
Assume normal condition constraints are satisfied. The word “flow” below refers only to post-contingency flow. How does SCOPF distinguish between the two cases in the following two situations:
Situation 1:• having one flow at 99% and • having one flow at 101%?
SCOPF
maxmin
maxmin
')(''
)(
0)(
hPhh
hPhh
Pg
)(min PfSubject to:
OBJECTIVE
Power Flow Eqts
“Normal Condition” constraints
Contingency constraints
Situation 2• having ten flows at 99% and one flow at 101% • having all flows below 50% except for one which is at 101%.
Basic Concepts3
Security level: A continuous function of operating conditions reflecting the “strength” of the power system with respect to a defined contingency set.
Risk level: A continuous function of operating conditions reflecting the “weakness” of the power system with respect to a defined contingency set.
Fact:(1) All “secure” operating conditions not equally secure.(2) All “insecure” operating conditions not equally risky.
Why? Because security level (or risk level) depends on(a)All flows (not just ones at the limits)(b)Contingency probabilities
Risk Evaluation4
N1,...,kcontgncies
),()Pr()( XESevEXRisk kk
Contingency Probabilities:• Always estimates• Reasonable default is proportional to line length• Can depend on line length, location, & weather, if outage data available.• Consider as weightings on severity reflecting contingency importance.
Severity function:
Post-contingency loading on each line
1
17
7
7
4
1
1
5
Risk Visualization
Severity circles/squares: represents a post-contingency violation or near-violation with the number corresponding to the violated circuit. Radial distance from the center of the diagram to each small circle is proportional to the extent (severity) of the violation.
Security regions : White center corresponds to loadings less than 90% of emergency rating.Yellow “doughnut” corresponds to loadings 90% -100%. Red outside corresponds to loadings in excess of emergency rating.
Probability sectors: sector angular spread is proportional to contingency probability
Illustration6
~ ~ ~
~
~
~
~
~
~ ~~
230kV
138kV
18 21 22
17
16 19 20
23
13
6
8
2 71
93 10
24 11 12
15
14
45
7
Model 2: (RBOPF)
Model 1: (SCOPF)
maxmin
maxmin
')(''
)(
0)(
hPhh
hPhh
Pg
)(min PfSubject to:
RMAXPRisk
hPhh
Pg
)(
)(
0)(
maxmin
)(min PfSubject to:
C4
C7
C21C3
17
7 C5
C6
7
4
1
1
Primary event
Level 1
Level 2
Level 3
Level 4
Level 5
2
1
3
1
4
7
5
7
6
7
7
4
2
1
3
1
0.038 0.005 0.005 0.020.29 0.02 0.4 0.05Level 1
Probability
SCOPF RBMO
CEI
0 0 1 100 0 0100 1
6 5
5.81
0
Severity
Cas
cadi
ng S
eque
nce
Stop Cascading Collapse
Illustration
8
RB-OPF: Visualization
All lines, 40 hrs, no contingency
All lines, 1 hr,40 contingencies
Vertical axis: angular separation across each line obtained from SCOPFHorizontal axis: angular separation across each line obtained from RBOPF
Points above the diagonal indicate lines for which SCOPF solution results in greater stress.Points below the diagonal indicate lines for which RBOPF solution results in greater stress.
9
Preventive RBOPF
0 0Min ( )f u
0
max0 0 0 0
max0
1 max
. . ( , ) 0 , 0,...,
( , )
( , ) , 1,...,
(Pr, ,..., ,..., )
k k
k k C k
k c R
s t g x u k c
h x u h
h x u K h k c
Risk x x x K Risk
10
Preventive-Corrective RBOPF
0 0
max0 0 0 0
max
0
max
Min ( )
. . ( , ) 0 , 0,...,
( , )
( , ) , 1,...,
| |
(Pr, ) , 0,...,
k k k
k k k C k
k
k R
f u
s t g x u k c
h x u h
h x u K h k c
u u u
Risk x K Risk k c
11
Preventive-Corrective RBOPF (4)
SCOPF RBOPF
Preventive RB-SCOPF Corrective RB-SCOPF
HSM(Kc=1)
ESM(Kc=1.05)
EESM(Kc=1.25)
HSM (Kc=1)
ESM(Kc=1.05)
EESM(Kc=1.25)
Risk 15.3 6.1 12.2 7.7 6.1 6.1 6.1 6.1
Cost ($) 1218909 1120194
1219067 1206506 1181047 1201542 1146556 1098027
CEI 102122 89222 101229 93912 102613 75564 63828 75749
ASI 3911 3336 3791 3347 3020 2817 3213 3457
12
RB-LMPs
k
tqtq
k
sqsq
kk P
F
P
F
P
LDLMP
1
k
uq
k
tq
k
sqq
kk P
FSev
P
FSev
P
FSevp
P
LRLMP 1
Deterministic:
Risk-based:
Loss costEnergy cost
Congest costLoss
component
Energy component Congest component for one contingency, (line s, t
above 100%, line u at 92%)
Control of risk level is uniformPrice signal for risk-relief is more effectiveLMPs are less volatile
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