Branch Outage Simulation for Contingency Studies

Dr.Aydogan OZDEMIR, Visiting Associate ProfessorDepartment of Electrical Engineering,

Texas A&M University, College Station TX 77843Tel : (979) 862 88 97 , Fax : (979) 845 62 59

E-mail : ozdemir@ee.tamu.edu

Aydoğan Özdemir 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 devicesand 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 SYSTEMSECURITY

monitoringcontingency analysissecurity 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 modelsand better computation algorithms

DC load flowsSensitivity factors

LINE OUTAGE SIMULATION

An outage of a line can either be simulated by setting its impedance, y ij = 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=0Sij=0i j

ji ji

Z-Matrix techniquesModification 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 0iy 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 T0..0]0...ΔP

1][,]00..1..0010..00[ BXXΔδ siPT

}/1{Re,/1[,/1[ ijyalijxk

ikxijx ii]B'ij]B',ΔδBΔP

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

silm

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

Involves more complicated models and better computation algorithms

Qij j i Qji

TijQ T

jiQ

LiQLj

Q

2sin]cos[}..{Im 022* i

ijiijjiijjijiijijiijb

VgVVbVVVyagQ VVCan be split up into two parts,

Transferring reactive powerassumed to flow through the line

Tij

Tji

jiijjiijjiTij

QQ

gVVbVVQ

sin2/][ 22

Loss reactive powerassumed to allocatedat the busses

LiLj

iji

ijjijijiLi

QQ

bVV

bVVVVQ

4

)(2

]cos2[ 02222

Line outage simulation by hypothetical reactive power sources

ji

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.

bijbus ibus 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 T

ijQ

LiQ LiQLi

Tijsi QQQ

0jiQ

1:a

ijbaa

)11

(1

TijQ T

jiQ

LiQLjQ 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

ijjk

ikk

kik*i QQVYVIm

Djsjj QQagQ

jiik

jkk

kjk*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

ijk xXXX

PP

/)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,

LjLi QQ~~

VBQVg )(

)()(

q

DjjijDiijiQwrt

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 [)()(,,( DjjijDiijiTij 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 maximumSingle direction search

SOLUTION BY GENETIC ALGORITHMSEvolutionary 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

optimizationcriteria

met

Generate initial population

evaluate objectivefunction

bestindividuals

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

ji

outaged branch

BASE CASE LOAD FLOW

SELECT AN OUTAGE

CALCULATE BUS VOLTAGE PHASE ANGLES

CALCULATE THE REMAINING QUANTITIES

END

QXVtosubject

Qwrt

QQMinimizeTij

jiij

NUMERICAL EXAMPLES IEEE 14-Bus test System

G

G

G

G

G

1

6

7

11 10 9

8

5 4

32

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 MvarQ5-6 = 12.42 MVar

Post-Outage Voltage Magnitudes for IEEE-14 Bus Test System

Outage of Line 7-9 Outage of transformer 5-6 Bus No VLF [pu] VPF [pu]

V [%] VLF [pu] VPF [pu] V [%]

1 1.060 1.060 0.0 1.060 1.060 0.0 2 1.045 1.045 0.0 1.045 1.045 0.0 3 1.010 1.010 0.0 1.010 1.010 0.0 4 1.015 1.015 0.0 1.015 1.023 0.8 5 1.016 1.018 0.2 1.025 1.032 0.7 6 1.070 1.070 0.0 1.070 1.070 0.0 7 1.066 1.068 0.1 1.055 1.055 0.0 8 1.090 1.090 0.0 1.090 1.090 0.0 9 0.988 0.993 0.5 1.046 1.038 0.8

10 0.994 0.999 0.5 1.043 1.036 0.7 11 1.027 1.030 0.3 1.053 1.049 0.4 12 1.050 1.051 0.1 1.052 1.054 0.2 13 1.040 1.041 0.1 1.049 1.048 0.1 14 0.992 0.996 0.4 1.028 1.024 0.4

Maximum error: 0.5 % Maximum error: 0.8 %

Line Outage of Line 7-9 Outage of transformer 5-6 l=m QPF

[MVar]

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.