Security-Constrained OPF
and
Risk-Based OPF
2
SCOPF
OBJECTIVE
min f ( P )
Subject to:
Power Flow Eqts
“Normal Condition” constraints
g (P)  0
h min  h ( P )  h max
Contingency constraints h ' min  h ' ( P )  h ' max
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%?
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 Concepts
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
3
Risk Evaluation
Risk ( X ) 
 Pr( E k )  Sev ( E k , X )
contgncies
k 1,..., N
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
4
Risk Visualization
C4
Security regions :
Probability sectors: sector
White center
angular spread is
corresponds to
proportional to
loadings less than
contingency probability
90% of emergency
1
7
1
rating.
C3
1 1
Yellow “doughnut”
C2
C5
7
corresponds to
C6
7
loadings 90% -100%.
4
Red outside
corresponds to
loadings in excess of
emergency rating.
Severity circles/squares: represents a post-contingency violation or
near-violation with the number corresponding to the violated circuit.
5
Radial distance from the center of the diagram to each small circle is
proportional to the extent (severity) of the violation.
C7
6
Illustration
230kV
18
~
21
~
22
~
17
23
~
~
16
19
20
14
13
~
15
~
~
24
11
3
12
10
9
5
4
8
138kV
1
~
6
2
~
7
~
Model 1:
(SCOPF)
Illustration
Model 2:
(RBOPF)
min f ( P)
min f ( P)
Subject to:
Subject to:
g ( P)  0
g ( P)  0
h min  h( P)  h max
h min  h( P )  h max
h'min  h' ( P)  h'max
7
Risk ( P )  RMAX
C4
C3
C2
7
1
Primary
event
2
3
4
5
6
7
2
3
Level 1
1
1
7
7
7
4
1
1
6
5
Level 2
Level 3
Level 4
Level 5
Level 1
Probability
C7
C5
C6
1
7

1
RBMO

1
7
Cascading Sequence
SCOPF
Severity
4
CEI
0.29
0.02
0.038
0.005
0.005
0.02
0.4
0
0
100
1
1
100
0
5.81
Stop Cascading
0.05
0

0
Collapse
RB-OPF: Visualization
All lines, 40 hrs, no contingency
All lines, 1 hr,40 contingencies
8
Vertical axis: angular separation across
each line obtained from SCOPF
Horizontal 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.
Preventive RBOPF
M in f 0 ( u 0 )
s .t . g k ( x k , u 0 )  0 , k  0, ..., c
h0 ( x 0 , u 0 )  h0
m ax
hk ( x k , u 0 )  K C  hk
m ax
, k  1, ..., c
R isk (P r, x1 , ..., x k , ..., x c )  K R  R isk m ax
9
Preventive-Corrective RBOPF
M in f 0 ( u 0 )
s .t . g k ( x k , u k )  0 , k  0, ..., c
h0 ( x 0 , u 0 )  h
m ax
0
hk ( x k , u k )  K C  hk
m ax
, k  1, ..., c
| u k  u 0 |  u
R isk (P r, x k )  K R  R isk m ax , k  0, ..., c
10
Preventive-Corrective RBOPF
11
(4)
Preventive RB-SCOPF
SCOPF RBOPF
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
112019
4
1219067
1206506
1181047
1201542
1146556
1098027
CEI
102122
89222
101229
93912
102613
75564
63828
75749
ASI
3911
3336
3791
3347
3020
2817
3213
3457
RB-LMPs
Deterministic:
DLMP
Energy
Energy
component
cost
Risk-based:
RLMP
k
k

L
   1 
 Pk

 F sq
 Ftq

   sq


tq


P
 Pk
k

LossLoss
component
cost

L
   1 
 Pk

Congest component for one
contingency, (line s, t above
Congest
100%,
linecost
u at 92%)
 
Control of risk level is uniform
Price signal for risk-relief is more effective
LMPs are less volatile
12
 

 Sev Ftq
 Sev Fuq

  Sev F sq
    pq  




 Pk
 Pk
 Pk





