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