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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 : [email protected]
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.
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.
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
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
N
SELECT A
NEW OUTAGE
LIMIT VIOLATION
Y
ALARM MESSAGE
N
LAST OUTAGE
Y
END
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
not feasible
for real-time
applications.
Check the limit
violations
approximate methods to quickly
identify conceivable contingencies
real-time applications
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
DC load flows
Sensitivity factors
voltage magnitude
violations
both
involves more complicated models
and better computation algorithms
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.
i
i
S ij  0
yi 0  0
yij  0
Sij=0
S ji  0
y j0  0
Z-Matrix techniques
Modification of ZBUS is
required for each outage
Sji=0
j
j
i S ij  0
S si
S ji  0j
yij
yi 0
y j0
S sj
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
SIMULATION FOR MW LINE FLOW PROBLEM
DC LOAD FLOW :
ΔP  BΔδ , [B']ij  1/ xij , [B']ii  1/ xik , xij  Re al{1/ yij }
k
outage of a line connected between busses i and j 
ΔP  [0 0.. 0 Psi 0... Psi 0..0]T ; Psi  Re al{S si }
Δδ  X[0 0 ..0 1 0 0.. 1..0 0]T Psi , X  [B ] 1
The new real power flow through the line connected between busses n and m can be
derived and approximated as,
1
~
Pnm  Pnm  Pnm  Pnm 
([X]nn  [X]mm - 2[X]nm ) Psi
xlm
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
i
Qij
QijT
Q Tji
Q Li
Qji
j
Q Lj
b
Qij   Im ag{Vi*. yij .V j }  [Vi2 ViV j cos  ji ] bij ViV j gij sin  ji Vi2 i0
2
Transferring reactive power
QijT  [Vi2  V j2 ]bij / 2  ViV j gij sin  ji
QTji  QijT
assumed to flow through
the line
Loss reactive power
QLi  [Vi2  V j2  2ViV j cos ji ]
QLj  QLi
bij
b
 (Vi2  V j2 ) i 0
2
4
assumed to allocated
at the busses
Can be split up
into two parts,
Line outage simulation by hypothetical reactive power sources
QijT
i Q 0
ij
Qsi  QijT  QLi
QLi
 QijT
QLi
Q ji  0 j
Qsi  QijT  QLi
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.
bus i
a :1
bus i
bij
bus j
1 1
( 1)bij
a a
QijT
QLi
bij
QTji
QLj
bus j
1
(1 )bij
a
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.
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,
*


Qi   Im ag V
Y
V
i  ik k    Q ik  Q ij  Q si  Q Di

 k j
k
*


Q j   Im ag V
Y
V

j
jk k    Q jk  Q ji  Qsj  Q Dj

 k i
k
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.
 l   l  ( X li  X lj ) Pk
Pk 
, l=2,3,…, NB
Pij
1  ( X ii  X jj  2 X ij ) / x k
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,
~
~
Q Li  Q Lj
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,
Minimize (Qi  Qij  Q Di ) ( Q j  Q ji  Q Dj )
T
wrt Qij
Subject to g q (V )  Q  BV  
~
Qij  QijT  QLi
~
Q ji  QijT  QLi
reactive power flows
through the outaged
line
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
L(QijT , V,   (Qi  Qij  QDi ) 2  ( Q j  Q ji  QDj ) 2    [B 1Q  V]
with respect to QijT , V and  . 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.
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.
Generate initial
population
evaluate objective
function
GENERATE NEW
POPULATION
selection
crossover
mutation
N
optimization
criteria
met
Y
best
individuals
result
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.
BASE CASE LOAD FLOW
SELECT AN OUTAGE
bounded network
CALCULATE BUS VOLTAGE PHASE ANGLES
i
j
outaged branch
Minimize Qij Q ji
wrt QijT
subject to V  X Q
CALCULATE THE
REMAINING QUANTITIES
END
NUMERICAL EXAMPLES
IEEE 14-Bus test System
G
G
3
2
G
1
5
4
G
8
7
G
6
11
10
9
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
12
13
14
Q7-9 = 27.24 Mvar
Q5-6 = 12.42 MVar
Post-Outage Voltage Magnitudes for IEEE-14 Bus Test System
Bus
Outage of Line 7-9
Outage of transformer 5-6
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 %
Post-outage reactive power flows for IEEE-14 Bus Test Systems
Line
l=m
1-2
Outage of Line 7-9
Outage of transformer 5-6
QPF
QDF
QPF
QDF
Q
Q
[MVa [Mvar
[MVar]
[Mvar]
[Mvar]
[Mvar]
r]
]
-20.3 -20.2
0.07 -21.6 -21.1
0.53
1-5
5.4
4.4
2-3
3.6
3.6
2-4
0.2
-0.1
2-5
2.8
1.7
3-4
5.3
5.0
4-5
12.0
9.0
4-7
-14.1
-14.8
4-9
13.2
12.9
5-6
12.8
13.8
6-11
14.6
12.9
6-12
3.7
3.5
6-13
13.0
12.0
7-9
0.98
0.02
0.27
1.15
0.33
3.02
0.70
0.32
0.97
1.73
0.20
0.96
86.7
9-10
-5.5
-4.8
9-14
-2.6
-1.9
10-11
-11.3
-10.2
12-13
1.9
1.6
13-14
8.3
7.4
7-8
-14.5
-13.3
0.71
0.70
1.11
0.34
0.85
1.21
1.3
3.3
-1.6
-1.3
3.7
8.6
-5.1
3.0
19.5
5.1
15.1
9.6
-8.2
-4.6
-14.9
3.4
12.4
-21.2
-1.3
3.3
-5.8
-4.2
-0.1
14.0
-0.8
6.4
42.6
19.9
4.7
15.5
17.7
-8.9
-5.5
-15.5
3.5
12.2
-21.2
2.64
0.03
4.15
2.90
3.81
5.35
4.31
3.35
0.41
0.36
0.42
8.12
0.66
0.88
0.64
0.06
0.24
0.04
IEEE 57-Bus Test System
5
G
4
3
G
G
2
1
16
2
G
45
18
6
19
15
17
14
13
20
21
12
G
46
47
44 48
26
50
24
49
23
38
22
2
39
37
25
40
56
57
41
11
36
27
28
7
29
35
30
33
31
32
34
53
54
52
42
43
55
8
G
G
9
10
51
First one is the outage of the line connected between bus-12 and bus-13, whose preoutage 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
V
Bus No Voltage magnitudes [p.u.]
pre-outage VPF
VDF
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
l-m Qlm Qml
1-2
1-15
3-15
50-51
75.00 -84.12
33.74 -23.95
-18.26 13.73
-4.16 6.51
QPF
Qlm
QDF
Qml
Qlm Qml
74.84 -83.94 75.01 84.14
45.29 -34.96 46.26 35.22
0.54 -5.15 0.87 -5.26
-9.43 9.92 -9.23 9.78
threshold error = 0.2 MVar.
Q
[MVar]
0.17
0.97
0.33
0.20
0.20
0.26
0.11
0.14
Post-Outage Voltage Magnitudes for outage of the transformer
connected between bus 13 and bus 49
Bus No
11
13
21
48
49
50
51
Voltage magnitudes [p.u.]
pre-outage
VPF
VDF
0.974
0.976
0.977
0.979
0.985
0.987
1.009
0.982
0.980
1.028
0.997
0.995
1.036
0.978
0.972
1.024
0.980
0.977
1.052
1.038
1.036
threshold error = 0.0015 p.u.
V
0.0011
0.0016
0.0017
0.0016
0.0056
0.0032
0.0018
Post-Outage Reactive Power Flows for outage of the transformer
connected between bus 12 and bus 13
Reactive Power Flow [MVar]
Line
l-m
3-15
12-13
15-45
14-46
47-48
48-49
50-51
10-51
pre-outage
Qlm
Qml
-18.26 13.73
60.27 -64.01
-0.79 2.15
27.32 -25.39
12.36 -12.26
-7.40 6.95
-6.16 6.51
12.47 -11.81
QPF
QDF
Qlm
Qml
Qlm
Qml
-15.59 11.01 -17.09 12.53
52.49 -56.76 50.06 -54.46
7.67 -5.67 9.33 -7.36
42.82 -39.29 45.93 -42.24
24.76 -24.41 22.71 -22.27
5.93 -6.10 4.31 -4.20
-13.25 14.53 -11.84 13.35
21.06 -19.83 23.24 -21.98
threshold error = 1.0 MVar.
Q
[MVar]
1.50
2.43
1.66
3.11
2.05
1.62
1.41
2.18
1.52
2.30
1.69
2.95
2.14
1.90
1.18
2.15