Branch Outage Simulation for

Contingency Studies

Dr.Aydogan OZDEMIR, Visiting Associate Professor

Department of Electrical Engineering,

Texas A&M University, College Station TX 77843

Tel : (979) 862 88 97 , Fax : (979) 845 62 59

E-mail : ozdemir@ee.tamu.edu

Aydoan zdemir was born in Artvin, Turkey, on

January 1957. He received the B.Sc., M.Sc. and

Ph.D. degrees in Electrical Engineering from

Istanbul Technical University, Istanbul, Turkey in

1980, 1982 and 1990, respectively. He is an

associate professor at the same University. His

current research interests are in the area of electric

power system with emphasis on reliability analysis,

modern tools (neural networks, fuzzy logic, genetic

algorithms etc.) for power system modeling,

analysis and control and high-voltage engineering.

He is a member of National Chamber of Turkish

Electrical Engineering and IEEE.

Outages of component(s)

Overstress on the other components

No limit violation limit violation(s)

operation of protective devices

and switching of the unit(s)

partial or total loss of load

Power System Security

Power system security is the ability of the system to withstand one or more component

outages with the minimal disruption of service or its quality.

POWER SYSTEM

SECURITY

monitoring

contingency analysis

security constrained opf

Monitoring : Data collection and state estimation

The objective of steady state contingency analysis is to

investigate the effects of generation and transmission

unit outages on MW line flows and bus voltage

magnitudes.

START

SET SYSTEM MODEL TO

INITIAL CONDITIONS

SIMULATE AN OUTAGE OF A

GENERATOR OR A BRANCH

LIMIT VIOLATION

Y

ALARM MESSAGE

LAST OUTAGE

Y

END

N

N

SELECT A

NEW OUTAGE

Real-time applications require fast and reliable computation methods due to the high number of

possible outages in a moderate power system.

However, there is a well-known conflict between the accuracy of the method applied and the

calculation speed.

Exact solution Full AC power flow

for each outage

Check the limit

violations

not feasible

for real-time

applications.

real-time applications

approximate methods to quickly

identify conceivable contingencies

AC power flows only for

critical contingencies.

Check the limit violations

APPROXIMATE CONTINGENCY ANALYSIS

Contingency ranking contingencies are ranked in an approximate order of a scalar performance

index, PI.

contingencies are tested beginning with the most severe one and

proceeding down to the less severe ones up to a threshold value.

Masking effect causes false orderings and misclassifications.

Contingency screening Explicit contingency screening is performed for all contingencies, following

an approximate solution (DC load flow, one iteration load flow, linear

distribution or sensitivity factors etc.)

Contingency screening is performed in the near vicinity of the outages (local

solutions)

Hybrid methods utilizing both the ranking and the screening

outage of a branch or a generation unit

MW line flow overloads voltage magnitude

violations

both

involves more complicated models

and better computation algorithms DC load flows

Sensitivity factors

LINE OUTAGE SIMULATION

An outage of a line can either be simulated by setting its impedance, yij = 0 or by injecting

hypothetical powers at both ends of the line. The latter method is preferred to preserve the

original base case bus admittance matrix.

Sji=0 Sij=0 i j

j i j i

Z-Matrix techniques

Modification of ZBUS is

required for each outage

Determination of the hypothetical sources so

that all the reactive power circulates through

the outaged line while maintaining the same

voltage magnitude changes in the system

0ijS 0ijy 0jiS

00 iy 00 jy

0ijSijy

0jiS

siS 0i

y 0jy sjS

SIMULATION FOR MW LINE FLOW PROBLEM

DC LOAD FLOW :

outage of a line connected between busses i and j

}{Re;0..00[ sisisisi SalPPP T

0..0]0...P

1][,]00..1..0010..00[

BXXsi

PT

}/1{Re,/1[,/1[ ijyalijx

kik

xij

x ii]B'

ij]B',BP

The new real power flow through the line connected between busses n and m can be

derived and approximated as,

si

lm

nmnmnmnm Px

PPPP )2[X]-[X]([X] nmmmnn 1~

See Power Generation, Operation and Control by Wood and Wollenberg for details

SIMULATION FOR VOLTAGE MAGNITUDE PROBLEM

Linear models are not sufficient for most outages

Reactive power flows can not be isolated from bus voltage phase angles

Involves more complicated models and better computation algorithms

Qij j i Qji

T

ijQ

T

jiQ

LiQ LjQ

2sin]cos[}..{Im

022* iijiijjiijjijiijijiij

bVgVVbVVVyagQ VV

Can be split up

into two parts,

Transferring reactive power assumed to flow through

the line Tij

Tji

jiijjiijjiTij

QQ

gVVbVVQ

sin2/][22

Loss reactive power assumed to allocated

at the busses

LiLj

iji

ij

jijijiLi

QQ

bVV

bVVVVQ

4

)(2

]cos2[02222

Line outage simulation by hypothetical reactive power sources

j i

For a tap changing transformer, cross flow through the equivalent impedance is considered to be the

transferring reactive power, where shunt flows can be considered as the loss reactive powers.

bij bus i bus j

bij

bus i bus j

Transferring reactive power is sensitive both to bus voltage magnitudes and bus voltage phase angles.

However, loss reactive power is dominantly determined by bus voltage phase angles and has a weak

coupling with bus voltage magnitudes. Therefore, transferring reactive powers are enough for a

reasonable accuracy.

0ijQ

LiTijsi

QQQ

TijQ

TijQ

LiQ LiQLi

Tijsi

QQQ

0jiQ

1:a

ijbaa

)11

(1

TijQ

TjiQ

LiQ

LjQ ijb

a)

11(

Hypothetical reactive power injections to bus i and bus j, will result in a change in net

reactive bus powers Qi and Qj. This in turn, will result in a change in system state

variables with respect to pre-outage values. This change must be equivalent to the

changes when the line is outaged.

Load bus reactive powers do not satisfy the nodal power balance equation due to the

errors in load bus voltage magnitudes calculated from linear models. Therefore, part

of the fictitious reactive generation flows through the neighboring paths instead

circulating through the outaged branch. These reactive power mismatches can

mathematically be expressed as,

Disii QQagQ

ij

jkik

kkik

*i QQVYVIm

Djsjj QQagQ

ji

ikjk

kkjk

*j QQVYVIm

where Qi and QDi are the net reactive power and the reactive demand at load bus i, is the

complex voltage at bus i and Yik is the element of bus admittance matrix. The superscript *

denotes the conjugate of a complex quantity. Calculated load bus voltage magnitudes need to

be modified in a way to minimize the bus reactive power mismatches at both ends of the

outaged line.

This can be accomplished a local optimization formulation

1. Select an outage of a branch, numbered k and connected between busses i and j.

2. Calculate bus voltage phase angles by using linearized MW flows.

kljlill PXX )(

kijjjii

ij

kxXXX

P

P/)2(1

, l=2,3,, NB

where X is the inverse of the bus suseptance matrix, Pij is the pre-outage active

power flow through the line and xk is the reactance of the line.

3. Calculate intermediate loss reactive powers,

4. Minimize reactive power mismatches at busses i and j, while satisfying linear reactive

power flow equations. Mathematically, this corresponds to a constrained optimization

process as,

LjLiQQ~~

VBQVg )(

)()(

q

DjjijDiiji

Qwrt

toSubject

QQQQQQMinimizeTij

reactive power flows

through the outaged

line

LiTijij QQQ

~

LiTijji QQQ

~

SOLUTION OF THE CONSTRAINED OPTIMIZATION PROBLEM

After having formulated the outage simulation as a constrained optimization problem,

minimization can be achieved by solution of the partial differential equations of the

augmented Lagrangian function

V]QBV

122[)()(,,( DjjijDiiji

Tij QQQQQQQL

with respect to . Note that V does not need to include all the load bus

voltage magnitudes; instead only busses i, j and their first order neighbors are enough

for optimization cycle.

andV,TijQ

Drawback : Convergence to local maximum

Single direction search

SOLUTION BY GENETIC ALGORITHMS Evolutionary algorithms are stochastic search methods that mimic the metaphor of natural biological

evolution.

Genetic Algorithms (GAs) are perhaps the most widely known types of evolutionary computation methods

today.

GAs operate on a population of potential solutions applying the principle of survival of the fittest procedure

better and better approximation to a solution. At each generation, a new set of better approximations is created

by selecting individuals according to their fitness in the problem domain. This process leads to the evolution

of populations of individuals that are better suited to their environment than the individuals that they were

created from.

Y

N

result

optimization

criteria

met

Generate initial

population

evaluate objective

function

best

individuals

GENERATE NEW

POPULATION

crossover

mutation

selection

For the details of the processes see

Cheng, Genetic

Algorithms&Engineering

Optimization by M. Gen, R., New

York: Wiley, 2000 . Such a single

population GA is powerful and

performs well on a broad class of

optimization problems.

bounded network

j i

outaged branch

BASE CASE LOAD FLOW

SELECT AN OUTAGE

CALCULATE BUS VOLTAGE PHASE ANGLES

CALCULATE THE

REMAINING QUANTITIES

END

QXVtosubject

Qwrt

QQMinimize

Tij

jiij

NUMERICAL EXAMPLES

IEEE 14-Bus test System

G

G

G

G

G

1

6

7

11 10 9

8

5 4

3 2

13

12

14

Base case control variables :

PG2 = 0.4 p.u.

PG3 = PG6 = PG8 = 0.0 p.u.

V1 = 1.06 p.u.

V2 = 1.045 p.u.

V3 = 1.01 p.u.

V6 = 1.07 p.u.

V8 = 1.09 p.u.

B9 = 0.19 p.u.

t4-7 = 0.978

t4-9 = 0.969

t5-6 = 0.932

Q7-9 = 27.24 Mvar

Q5-6 = 12.42 MVar

P o s t -O u ta g e V o lta g e M a g n itu d e s fo r I E E E -1 4 B u s T e s t S y s te m

O u ta g e o f L in e 7 -9 O u ta g e o f tra n s fo rm e r 5 -6 B u s

N o V L F [p u ] V P F [p u ] V [% ] V L F [p u ] V P F [p u ] V [% ]

1 1 .0 6 0 1 .0 6 0 0 .0 1 .0 6 0 1 .0 6 0 0 .0

2 1 .0 4 5 1 .0 4 5 0 .0 1 .0 4 5 1 .0 4 5 0 .0

3 1 .0 1 0 1 .0 1 0 0 .0 1 .0 1 0 1 .0 1 0 0 .0

4 1 .0 1 5 1 .0 1 5 0 .0 1 .0 1 5 1 .0 2 3 0 .8

5 1 .0 1 6 1 .0 1 8 0 .2 1 .0 2 5 1 .0 3 2 0 .7

6 1 .0 7 0 1 .0 7 0 0 .0 1 .0 7 0 1 .0 7 0 0 .0

7 1 .0 6 6 1 .0 6 8 0 .1 1 .0 5 5 1 .0 5 5 0 .0

8 1 .0 9 0 1 .0 9 0 0 .0 1 .0 9 0 1 .0 9 0 0 .0

9 0 .9 8 8 0 .9 9 3 0 .5 1 .0 4 6 1 .0 3 8 0 .8

1 0 0 .9 9 4 0 .9 9 9 0 .5 1 .0 4 3 1 .0 3 6 0 .7

1 1 1 .0 2 7 1 .0 3 0 0 .3 1 .0 5 3 1 .0 4 9 0 .4

1 2 1 .0 5 0 1 .0 5 1 0 .1 1 .0 5 2 1 .0 5 4 0 .2

1 3 1 .0 4 0 1 .0 4 1 0 .1 1 .0 4 9 1 .0 4 8 0 .1

1 4 0 .9 9 2 0 .9 9 6 0 .4 1 .0 2 8 1 .0 2 4 0 .4

M a x im u m e r ro r : 0 .5 % M a x im u m e r ro r : 0 .8 %

Line Outage of Line 7-9 Outage of transformer 5-6

l=m QPF

[MVa

r]

QDF

[Mvar

]

Q

[Mvar]

QPF

[MVar]

QDF

[Mvar]

Q

[Mvar]

1-2 -20.3 -20.2 0.07 -21.6 -21.1 0.53

1-5 5.4 4.4 0.98 1.3 -1.3 2.64

2-3 3.6 3.6 0.02 3.3 3.3 0.03

2-4 0.2 -0.1 0.27 -1.6 -5.8 4.15

2-5 2.8 1.7 1.15 -1.3 -4.2 2.90

3-4 5.3 5.0 0.33 3.7 -0.1 3.81

4-5 12.0 9.0 3.02 8.6 14.0 5.35

4-7 -14.1 -14.8 0.70 -5.1 -0.8 4.31

4-9 13.2 12.9 0.32 3.0 6.4 3.35

5-6 12.8 13.8 0.97 42.6

6-11 14.6 12.9 1.73 19.5 19.9 0.41

6-12 3.7 3.5 0.20 5.1 4.7 0.36

6-13 13.0 12.0 0.96 15.1 15.5 0.42

7-9 86.7 9.6 17.7 8.12

9-10 -5.5 -4.8 0.71 -8.2 -8.9 0.66

9-14 -2.6 -1.9 0.70 -4.6 -5.5 0.88

10-11 -11.3 -10.2 1.11 -14.9 -15.5 0.64

12-13 1.9 1.6 0.34 3.4 3.5 0.06

13-14 8.3 7.4 0.85 12.4 12.2 0.24

7-8 -14.5 -13.3 1.21 -21.2 -21.2 0.04

Post-outage reactive power flows for IEEE-14 Bus Test Systems

G GG

G

G

G G

5

17

30

25

5429 5352

27

28

26 24

21

23 22

201918

5110

7

8 9

1234

6

35

34

33

3231

38

37

36

14 13 12

15

16

46

44

45

49

48

47

50

40

5739

55

41

42

56 11

43

2

2

IEEE 57-Bus Test System

First one is the outage of the line connected between bus-12 and bus-13, whose pre-

outage reactive power flow is 60.27 Mvar. Second case is the outage of a transformer

with turns ratio 0.895 connected between bus-13 and bus-49, whose pre-outage reactive

power flows is 33.7 Mvar.

Post-Outage Voltage Magnitudes for outage of the line connected between bus 12 and bus

Voltage magnitudes [p.u.] Bus No pre-outage VPF VDF

V

13 0.979 0.955 0.953 0.0019

14 0.970 0.953 0.951 0.0018

20 0.964 0.955 0.953 0.0016

46 1.060 1.042 1.040 0.0023

47 1.033 1.016 1.014 0.0016

48 1.028 1.011 1.009 0.0020

49 1.036 1.019 1.017 0.0024

threshold error = 0.0015 p.u.

Post-Outage Reactive Power Flows for outage of the

line connected between bus 12 and bus 13

Reactive Power Flow [MVar]

Line pre-outage QPF QDF

l-m Qlm Qml Qlm Qml Qlm Qml

Q

[MVar]

1-2 75.00 -84.12 74.84 -83.94 75.01 84.14 0.17 0.20

1-15 33.74 -23.95 45.29 -34.96 46.26 35.22 0.97 0.26

3-15 -18.26 13.73 0.54 -5.15 0.87 -5.26 0.33 0.11

50-51 -4.16 6.51 -9.43 9.92 -9.23 9.78 0.20 0.14

threshold error = 0.2 MVar.

Post-Outage Voltage Magnitudes for outage of the transformer

connected between bus 13 and bus 49

Voltage magnitudes [p.u.] Bus No pre-outage VPF VDF

V

11 0.974 0.976 0.977 0.0011

13 0.979 0.985 0.987 0.0016

21 1.009 0.982 0.980 0.0017

48 1.028 0.997 0.995 0.0016

49 1.036 0.978 0.972 0.0056

50 1.024 0.980 0.977 0.0032

51 1.052 1.038 1.036 0.0018

threshold error = 0.0015 p.u.

Post-Outage Reactive Power Flows for outage of the transformer

connected between bus 12 and bus 13

Reactive Power Flow [MVar]

Line pre-outage QPF QDF

l-m Qlm Qml Qlm Qml Qlm Qml

Q

[MVar]

3-15 -18.26 13.73 -15.59 11.01 -17.09 12.53 1.50 1.52

12-13 60.27 -64.01 52.49 -56.76 50.06 -54.46 2.43 2.30

15-45 -0.79 2.15 7.67 -5.67 9.33 -7.36 1.66 1.69

14-46 27.32 -25.39 42.82 -39.29 45.93 -42.24 3.11 2.95

47-48 12.36 -12.26 24.76 -24.41 22.71 -22.27 2.05 2.14

48-49 -7.40 6.95 5.93 -6.10 4.31 -4.20 1.62 1.90

50-51 -6.16 6.51 -13.25 14.53 -11.84 13.35 1.41 1.18

10-51 12.47 -11.81 21.06 -19.83 23.24 -21.98 2.18 2.15

threshold error = 1.0 MVar.