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Master Thesis
on
―Sizing and Location of Shunt FACTS
Devices inPower System using Genetic
Algorithms‖
Thesis Advisor: MSc and PhD Mario Rios
A Dissertation presented to the Faculty of the Electrical Engineeringof
Universidad de los Andesin Partial Fulfillment of the Requirements for the
Degree ofMaster of Electrical Engineering
by
Cesar Rodríguez
June 2011
2
2011 Cesar Rodríguez
ALL RIGHTS RESERVED
3
Sizing and Location of Shunt FACTS Devices in
Power System using Genetic Algorithms
Cesar Rodríguez
Universidad de los Andes
Abstract
FACTS (Flexible AC Transmission Systems) devices are used to improve
different problems in a power system. In this work, a Genetic Algorithm (GA)
is used to solve a multi-objective optimization problem which consists of
finding the optimal size and location of the shunt FACTS devices. The
objective is to enhance both voltage stability margin and first swing stability
margin in a multi-machine power system. The analysis is done for important
contingencies that cause problems both for voltage stability margin and first
swing stability margin. In this new method the location and sizing of shunt
FACTS devices are optimized simultaneously. Two different kinds of shunt
FACTS devices are used for transient and steady-state studies: Static Var
Compensator (SVC) and Static Synchronous Compensator (STATCOM). The
proposed method is evaluated on a 68 bus test system. The results obtained
show that proposed algorithm can find the optimal location and sizing of shunt
FACTS devices in a power system.
4
ACKNOWLEDGEMENTS
It is a pleasure to thank the many people who made this thesis possible.
This work would not have been possible without the support from my advisor
Prof. Dr. Mario Rios under whose guidance, I chose this topic.
I would like to say a big thanks to Mr. Oscar Gomez and Mr. Andres Leal,
who support was important in this work.
I want to say thank to Katherine Ovalle, Camilo Ordonez and Carlos Silva,
whose contributions were important in the organization and presentation of the
thesis.
My final words go to my family. I want to thank my family, whose love and
guidance is with me in whatever I pursue.
5
TABLE OF CONTENTS
1. Introduction ................................................................................................................................. 9
1.1. General .................................................................................................................................... 9
1.1. Shunt Flexible AC Transmission Systems (FACTS) ............................................................ 10
1.2. State of the Art ...................................................................................................................... 13
1.2.1. Optimal Location and Sizing of FACTS Devices for Voltage Stability and for First Swing
Stability 13
1.2.2. Genetic Algorithms for Optimal Location and Sizing of FACTS Devices ....................... 14
1.3. Motivation ............................................................................................................................. 15
1.4. Thesis Organization............................................................................................................... 15
2. Voltage Stability Index for Location and Sizing of Shunt FACTS Devices ............................. 17
2.1. Introduction ........................................................................................................................... 17
2.2. Index used to allocate and size the shunt FACTS devices .................................................... 17
2.3. Critical Contingencies for Voltage Stability or Loading Margin .......................................... 18
2.4. Simulation Results................................................................................................................. 18
2.5. Conclusions ........................................................................................................................... 23
3. First Swing Stability Index for Location and Sizing of Shunt FACTS Devices ....................... 24
3.1. Introduction ........................................................................................................................... 24
3.2. Index used to allocate and size the shunt FACTS devices .................................................... 24
3.3. Critical Contingencies for First Swing Stability Margin ....................................................... 27
3.4. Simulation Results................................................................................................................. 27
3.5. Conclusions ........................................................................................................................... 30
4. Optimal Location and Sizing of Shunt FACTS Devices using Genetic Algorithm (GA) ......... 31
4.1. Introduction ........................................................................................................................... 31
4.2. Multi-Objective Formulation ................................................................................................ 31
4.2.1. Voltage Stability Margin or Loading Margin Function .................................................... 32
4.2.2. First Swing Stability Margin Function .............................................................................. 32
4.2.3. Problem Constrains ........................................................................................................... 32
4.2.4. Critical Contingency ......................................................................................................... 33
4.2.5. Multi-objective Optimization Problem ............................................................................. 33
4.3. Genetic Algorithms and its Implementation ......................................................................... 34
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4.3.1. Individual Configuration ................................................................................................... 34
4.3.2. Initial Population Configuration ........................................................................................ 35
4.3.3. Fitness Function ................................................................................................................ 35
4.3.4. Reproduction Operator ...................................................................................................... 36
4.3.5. Crossover Operator ........................................................................................................... 36
4.3.6. Mutation operator .............................................................................................................. 37
4.4. Testing Results ...................................................................................................................... 38
4.4.1. Critical Contingencies Analyzed ....................................................................................... 39
4.4.2. GA Solution ...................................................................................................................... 39
4.4.3. Application of the GA Solution ........................................................................................ 41
4.5. Conclusions ........................................................................................................................... 45
5. General Conclusions ................................................................................................................. 46
7
LIST OF TABLES
Table I. Eigen-value for base case .................................................................................................... 19
Table II. Participation factor for the λmin (JR) .................................................................................. 19
Table III. System Loading for SVC and STATCOM for different Locations .................................. 20
Table IV. Size of SVC and STATCOM for different Locations on the System .............................. 21
Table V. Best Location of Shunt FACTS Devices for Critical Contingencies ................................. 22
Table VII. System instability margin for the critical contingencies on the system ........................... 28
Table VIII. Critical Contingencies for Voltage and Angle Stability ................................................. 39
Table IX. Optimal Location and Sizing for Each Shunt Facts Device .............................................. 40
Table X. Improved Contingencies for Both Devices ........................................................................ 40
Table XI. Objective and Fitness Function Value for Each Device ................................................... 41
8
LIST OF FIGURES
Figure 1. STATCOM Controller ..................................................................................................... 11
Figure 2. SVC controller ................................................................................................................. 11
Figure 3. V-I characteristic of the (a) STATCOM and of the (b) SVC ........................................... 12
Figure 4. Improvement of transient stability using (a) STATCOM and (b) SVC ............................ 13
Figure 5. Single line diagram of 68 bus test system. ....................................................................... 18
Figure 6. Participation factor and system loading for different location of SVC and STATCOM ... 20
Figure 7. Schematic for determine the size of shunt FACTS device ................................................ 21
Figure 8. Critical contingencies for voltage stability margin ........................................................... 22
Figure 9. Schematic of a machine speed and decelerating power during a fault on the system........ 25
Figure 10. Unstable situation and too unstable situation. ................................................................ 26
Figure 11. Critical contingencies for first swing stability margin .................................................... 27
Figure 12. Best locations of shunt FACTS devices to improve the FSS margin under contingencies
on the line 22 and line 49 .................................................................................................................. 28
Figure 13. Best locations of shunt FACTS devices to improve the FSS margin under contingencies
on the line 18 and line 86 .................................................................................................................. 29
Figure 14. Fault on the line 41 on the system. ................................................................................ 29
Figure 15. Individual configuration of shunt FACTS device. ......................................................... 34
Figure 16. Initial population configuration of shunt FACTS device. .............................................. 35
Figure 17. Roulette wheel reproduction operator. ........................................................................... 36
Figure 18. Single-point crossover point .......................................................................................... 37
Figure 19. Single-bit point mutation operator. ................................................................................ 37
Figure 20. Flowchart of GA implemented. ..................................................................................... 38
Figure 21. Fitness functions during optimization process. .............................................................. 41
Figure 22.System loading with SVC. .............................................................................................. 42
Figure 23.System loading with STATCOM.................................................................................... 42
Figure 24. Machine angles during line faults without shunt FACTS devices. ................................ 43
Figure 25. Machine angles during line faults with SVC. ................................................................ 44
Figure 26. Machine angles during line faults with STATCOM. ..................................................... 44
9
Chapther 1
1. Introduction
1.1. General
The electrical power systems, with the increase of the load and the
Deregulated Market, have been changing their way to be operated. This has
made that planning engineers look more interesting in including different
devices into system in order to counter the technical problems that it
brings[18].
One such device, used in power systems to solve different problems, is the
Flexible AC Transmission Systems (FACTS).
The use of FACTS devices in an electrical power system provide important
technical benefits such as: load flow control, voltage control, transient
stability, dynamic stability, system loadability, etc. There are different kinds
of FACTS devices: shunt connected controller, series connected controller and
combined shunt and series connected controller. Some parameters and
variables of a power system that can be controlled by FACTS devices in a fast
and effective way are: line impedance, terminal voltage, and voltage angle [1].
The main objectives of the shunt FACTS devices consist of increasing the
steady-state transmittable power and controlling the voltage profile by
appropriate reactive shunt compensation; in addition, these can also improve
the transient stability limit and damp power oscillations [1]. However, by
doing an optimal location and sizing of the shunt FACTS devices, these
technical benefits can be enhanced even more.
In this thesis, first, the sizing and location of shunt FACTS devices to improve
the system loading margin or voltage stability margin is found. Second, the
10
sizing and location of shunt FACTS devices to improve the first swing
stability margin is found. Finally, a multi-objective genetic algorithm to obtain
the optimal sizing and location of shunt FACTS devices to improve both the
first swing stability margin and voltage stability margin is developed.
The first swing stability margin is determined by analyzing the output results
of the conventional Time Domain Simulation (TDS) method, basically,
machine angles and speeds [12]. On the other hand, the voltage stability
margin or loading margin is calculated using Continuous Power Flow (CPF).
The analysis is done for important contingencies that cause problems both for
voltage stability and first swing stability margin.
1.1. Shunt Flexible AC Transmission Systems (FACTS)
In this section the main characteristics of shunt FACTS devices, Static Var
Compensator (SVC) and Static Synchronous Compensator (STATCOM), are
presented and a comparison between these two devices is done.
The use of FACTS devices in an electrical power system provide important
technical benefits such as: load flow control, voltage control, transient
stability, dynamic stability, system loadability, etc. There are different kinds
of FACTS devices: shunt connected controller, series connected controller and
combined shunt and series connected controller. Some parameters and
variables of a power system that can be controlled by FACTS devices in a fast
and effective way are: line impedance, terminal voltage, and voltage angle [1].
The main objectives of the shunt FACTS devices consist of increasing the
steady-state transmittable power and controlling the voltage profile by
appropriate reactive shunt compensation; in addition, these can also improve
the transient stability limit and damp power oscillations [1].
Static Synchronous Compensator (STATCOM):It is a static synchronous
generator operated as a static var compensator connected in parallel as
11
inductive or capacitive output current can be controlled independent of the
voltage system.
Figure1. STATCOM Controller
The STATCOM is one of the main FACTS controllers. It can be based
on voltage source converter or power supply. Figure1, shows a
schematic diagram of a STATCOM on voltage source converter and current
source converter.
Static var Compensator (SVC): The SVC can generate o absorber reactive
power and it’s connected in parallel as output is adjusted to interchange
inductive or capacitive current to maintain electrical power system specific
parameters (usually bus voltage).
Figure2. SVC controller
This overall isa thyristor switched reactor and /or a thyristor-
controlled capacitor or a combination, see Figure2.
12
Comparison between SVC and STATCOM
The SVC maximum var output decreases with the square of the AC system
voltage, therefore the support of reactive power during low system voltage is
limited. On the other hand, the STATCOM can be operated over its full output
current range even at very low system voltage levels. This basic operational
difference makes the STATCOM better than the SVC in order to increase the
system loading margin, since if the system load is increasing the busbars
voltage decrease and during this process it’s necessary to give reactive power
support [1]. On the other hand, this characteristic makes the STATCOM more
effective than the SVC to improve the first swing stability margin by
increasing the transmittable power. This concept can be seen in the equal areas
criterion, where the transmittable power is increased in the post-fault period
and the decelerating area can be bigger than accelerating area and therefore
the first swing stability margin increases. This comparison is made assuming
that the rating of both devices is the same. The Figure 3 and Figure4, show in
graphical form the two characteristic of the shunt FACTS device mentioned
above. These show that the STATCOM can be operated on its range of output
current even atvery low (theoretically zero), generally about 0.2 p.u., voltage
levels of the system[1].
Figure 3. V-I characteristic of the (a) STATCOM and of the (b) SVC
13
Figure4. Improvement of transient stability using (a) STATCOM and (b) SVC
The increase in stability margin obtainable
with the STACOM on a conventional thyristor-controlled SVC of equal
status is clearly illustrated by the use of equal-area criterion (see Figure4)[1].
1.2. State of the Art
In this section a review of the state of the art is presented. This review is done
based on works which optimal location and sizing of FACTS devices are
found to improve the voltage stability or system loading and first swing
stability. The use of GA to find both size and location of FACTS devices is
shown also.
1.2.1. Optimal Location and Sizing of FACTS Devices for
Voltage Stability and for First Swing Stability
The FACTS devices are important elements on the electrical power system in
order to improve many problems on as system. However, if these controllers
14
are allocated and sized optimally their performance is better and the problems
are improved even more.
Many researchers have worked on the location and sizing of FACTS devices
to improve different problems on power system. Usually, the location and size
is selected using different methods or indices. In [2], [3], a sensitivity of
system loading factor with respect to reactive power generation based
approach and the L-index of load buses are used, respectively. Both the factor
and the L-index are used to determine the optimal location of SVC for voltage
stability enhancement.Using non-dominated sorting particle swarm
optimization (NSPSO) for optimal locating multi-type FACTS devices in
order to optimize multi-objective voltage stability problem is presented in [4].
The enhancement of system static security and to enhance the system
performance under network contingencies through an optimal placement and
optimal setting of static var compensator (SVC) is presented in [6]. The
selection of optimal location and setting is based on single contingency
voltage sensitivity (SCVS) index.
On the other hand, the literature about the location and sizing of FACTS
devices to improve the first swing stability margin is more limited.In [7], the
first swing stability marginis improved by placing optimally a SVC using a
location index, as shown. The location index is based on the concept of
transient energy function method; however, the size in that work is not
determined.
1.2.2. Genetic Algorithms for Optimal Location and Sizing of
FACTS Devices
The power system can present different problems simultaneously. Therefore,
other authors have also solved multi-objective optimization problems by
finding the size and location of FACTS devices. In recent years, a method
used to solve these optimization problems is Genetic Algorithm (GA).
15
In [8], the optimal choice and allocation of FACTS devices in multi-machine
power system using genetic algorithm to achieve the power system economic
generation and dispatch.
In [9], a GA to improve the branch loading and voltage level is used,
optimizing the type, size and location of FACTS devices. The voltage stability
and voltage deviation are enhanced by finding both size and allocation of
different FACTS devices, as shown in [10]. The transient stability
performance is achieved determining the optimal location of shunt FACTS
devices for a long transmission line with predefined direction of real power
flow. This method is evaluated on a two area system.
1.3. Motivation
In this paper, a new method to find the optimal location and sizing of shunt
FACTS devices to optimize a multi-objective problem is proposed. The multi-
objective function, which is formed by first swing stability margin and voltage
stability margin, is maximized using GA. The analysis is done for important
contingencies that cause problems both for voltage stability and first swing
stability margin.
1.4. Thesis Organization
The remainder of this thesis is organized as follows. In Chapther2, the location
and sizing of the SVC and the STATCOM is found in order to improve the
voltage stability margin or loading system. The participation factor of the
system busbars of Jacobian matrix is used to allocate the controllers. Equally,
the location and size of shunt FACTS devices is determined to enhance the
first swing stability margin is presented in the Chapther 3. The Stability
Margin (SM) and Instability Margin (IM) of the system are used to allocate
the devices.
In the Chapther 4, a GA is implemented in order to find the allocation and
sizing of shunt FACTS devices for a multi-objective problem. The multi-
16
objective function, which is formed by first swing stability margin and voltage
stability margin, is maximized. Finally, conclusions are summarized in
Chapther 5.
17
Chapther 2
2. Voltage Stability Index for Location and Sizing
of Shunt FACTS Devices
2.1. Introduction
The loading margin, for a particular operating point, is the amount of
additional load, both active and reactive power, in a specific pattern of load
increase that would cause a voltage collapse [13]. Therefore, the loading
margin of a power system is an important measure of its proximity to voltage
collapse, and by increasing it; the voltage stability margin is also increased.
2.2. Index used to allocate and size the shunt FACTS
devices
―Although modal analysis is a general mathematic concept in which system
decomposition is achieved by eigen-analysis, here we use the term to refer
specifically to the analysis technique for voltage stability assessment using
eigen-analysis of the reduced steady state Jacobian matrix (JR) of a power
system‖ [14].
After determine the modal analysis the eigen-analysis of JR can be
summarized follows[14]:
o If all the eigenvalues are positive, the system is voltage stable.
o If at least one eigenvalue is equal to zero, the system is on the verge of
voltage instability.
o If any of the eigenvalues are negative, thee system is voltage unstable.
18
o Bus, branch, and generator participations are calculated based on the
right and left eigenvectors. The participations define the degree to
which each of these elements is associated with a particular mode.
In this work, first, the minimumEigen-value of JR (λmin(JR)) is calculated.
Second, the bus participation factors for λmin are determined. Finally, the best
location both SVC and STATCOM corresponds to the buses with participation
factor higher.
2.3. Critical Contingencies for Voltage Stability or Loading Margin
The critical contingencies analyzed are those that cause voltage instability in
the power system. In this case, a contingency is considered critical in voltage
if the system loading under that contingency is 0.
2.4. Simulation Results
The studied method is validated on a 68 bus test system, which is fully
explained in [17]. The single line diagram of the test system is shown in Fig.
5.
G07
07
23 05G04
04
G05
19
20
G06
06
22
68
21
65
62
63
G03
03
64
66
67
58
G02
02
60
59
57
56
52
37
27
24G09
29
09
28
2625
G08
08
54
G01
01
55
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
15
42
G14
14
41
NETS NYPSAREA#5
AR
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#4
AREA# 3
Figure 5. Single line diagram of 68 bus test system.
19
The Table I shows a list of the most critical buses on the system with their
respective eigen-value. This list is obtained of the modal analysis result and
shows that the bus most critical is the bus 40 and its eigen-values is 8.222
which is the λmin (JR). The base case corresponds to system without outage
lines.
Table I. Eigen-value for base case
Bus λ(JR)
40 8.222
64 10.801
39 12.025
49 13.864
50 27.111
With the information obtained in the Table I, now the participation factors are
calculated for the most critical bus (bus 40). In Table II is shown the
participation factor for the most critical bus.
Table II. Participation factor for the λmin (JR)
Bus Participation Factor
40 0.1174
48 0.1084
49 0.0993
46 0.0829
47 0.0787
39 0.0447
38 0.0360
51 0.0320
45 0.0299
With these factors, the best location of shunt FACTS devices is found. The
Figure 6 shows the participation factors for λmin and the location of SVC and
STATCOM in different buses of the system. As seen the best location
corresponds to the bus with higher participation factor, the bus 40 both SVC
and STATCOM.
20
Figure 6. Participation factor and system loading for different location of SVC and
STATCOM
The Table III shows the system loading for both devices. In general with the
STATCOM the system presents a higher loading.
Table III. System Loading for SVC and STATCOM for different Locations
Bus SVC STATCOM
40 1.3956 1.3956
50 1.3890 1.3890
51 1.3854 1.3854
48 1.3843 1.3843
49 1.3819 1.3819
On the other hand, the sizing of shunt FACTS device is determine as the value
of reactive power in the noise point when the system load increase until the
collapse point. The Figure 7 shows the behavior of a system bus during the
CPF analysis.
30 35 40 45 50 55
0.02
0.04
0.06
0.08
0.1
Bus
Pa
rtic
ipa
tio
n f
act
or
30 35 40 45 50 551.35
1.36
1.37
1.38
1.39
Bus
Lo
ad
ing
[
]
SVC
STATCOM
21
Figure 7. Schematic for determine the size of shunt FACTS device
In Table IV, the size of SVC and STATCOM is presented for the same
locations.
Table IV. Size of SVC and STATCOM for different Locations on the System
Bus SVC STATCOM
40 162.0 178.9
50 139.5 173.1
51 141.4 172.1
48 147.9 173.5
49 118.0 132.4
The location can be determined when the presents outage lines. In this case
some critical contingencies are considered to the analysis. The
Figure 8 shows the single line diagram of the system with the critical
contingencies for the voltage stability margin or system loading.
0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.5
-1
-0.5
0
0.5
1
1.5
2
Loading Parameter (p.u.)
VBus
qFACTS
Vm
qmax
22
G07
07
23 05G04
04
G05
19
20
G06
06
22
68
21
65
62
63
G03
03
64
66
67
58
G02
02
60
59
57
56
52
37
27
24G09
29
09
28
2625
G08
08
54
G01
01
55
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
15
42
G14
14
41
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Unstable
Figure 8. Critical contingencies for voltage stability margin
The best location of shunt FACTS devices under contingencies on the system
is presented inTable V. This location is based in the modal analysis results,
identifying the bus with the minimum eigen-value in the system.
Table V. Best Location of Shunt FACTS Devices for Critical Contingencies
Outage Line From Bus To Bus Bus
48 40 41 50
49 40 48 49
63 45 51 47
86 18 50 50
23
2.5. Conclusions
Bus SVC STATCOM The location and sizing for shunt FACTS devices is presented in order
to improve the voltage stability margin.
The location is determined with the modal analysis results. The bus with higher participation factor is the best location for both
devices.
The size is determined as the value of reactive power in the noise point
when the system load increase until the collapse point.
The STATCOM is most effective in order to increase the system
loading.
24
Chapther 3
3. First Swing Stability Index for Location and
Sizing of Shunt FACTS Devices
3.1. Introduction
Improvement of first swing stability (FSS) limit is one of the major concerns
in power system operation and planning studies [7]. The FSS is evaluated after
a perturbation on the system and it can be consider stable or unstable. The
stability of the system in the post-fault period is evaluated with the machine
angles.A system is stable in first-swing if the machine angles are bounded and
thus the speeds change sign in the post-fault period. In other words, a system
is considered to be stable if all the machine angles are lower than 180° in the
center of inertia (COI) reference frame [12], [15]. Also, a first swing stable
system can be considered as stable if the system has adequate damping in
subsequent swings [7].
On the other hand, a system is considered to be unstable if the angle of at least
one of the machines becomes unbounded. In other words, a system is
considered to be unstable if the angle of at least one of the machines is higher
than 180° in the COI reference frame [12].
Therefore, improve the first swing stability margin is important to maintain
the system security.
3.2. Index used to allocate and size the shunt FACTS
devices
In this section two indices are used to allocate the shunt FACTS devices.
These indices depends if the system is stable or not.
25
When the system is stable the first swing stability margin of the jth machine
(SMj) can be considered as:
Where Pdj (tpj) is the decelerating power of the jth machine at time tpj when its
speed changes sign or becomes zero, and refers to the maximum
decelerating power of the machine found between the clearing time (tc) and
tpj[12] and S is the set of Severely Disturbed Machines (SDMs).Figure 9
shows the behavior of the machine speed and decelerating power during a
fault on the system.
Figure 9. Schematic of a machine speed and decelerating power during a fault on the
system
Therefore, the first swing stability margin (SM) of the system can be
expressed as:
SM = min (SMj)
The best location of shunt FACTS devices is as that SM is maximized.
0 0.5 1 1.5-1
0
1
Dec
eler
ati
ng p
ow
er [
p.u
.]
Time [s]
0 0.5 1 1.5-1
0
1
Sp
eed
[p
.u.]
tp
Pd(t
p)
Pmax
d
26
When the system is unstable, the first swing instability margin of the jth
machine (IMj) can be expressed as
Where Mj is the inertia constant of the jth machine and is the minimum
post-fault speed of the jth machine. U is the set of unstable machines on the
system.
The value depends if the system is unstable or too unstable. The Figure
10 shows both situations in the system and the value consider for each
situation.
Figure 10. Unstable situation and too unstable situation.
Therefore, the first swing instability margin (IM) of the system can be
expressed as:
The best location of shunt FACTS devices is as that the system pass from
unstable situation to stable situation.
0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
Time [s]
Sp
eed
[p
.u.]
Unstable
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
10
20
30
40
50
Time [s]
Sp
eed
[p
.u.]
Toounstable
tcl
27
3.3. Critical Contingencies for First Swing Stability
Margin
The critical contingencies analyzed are those that cause angular instability in
the power system. A contingency is considered critical in angle if the angle of
at least one of the machines is higher than 180° or if the system has not
adequate damping in subsequent swings.
3.4. Simulation Results
The system analyzed is the 68 bus test system and the critical contingencies
forfirst swing stability margin.
G07
07
23 05G04
04
G05
19
20
G06
06
22
68
21
65
62
63
G03
03
64
66
67
58
G02
02
60
59
57
56
52
37
27
24G09
29
09
28
2625
G08
08
54
G01
01
55
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
42
G14
14
41
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#4
AREA# 3
Unstable
Figure 11. Critical contingencies for first swing stability margin
The Table VI shows system instability margin for the critical contingencies on
the system.
28
Table VI. System instability margin for the critical contingencies on the system
Outage Line SDMs IMj IM
21-22 6 0.5002
0.5002 7 0.2962
40-41 11 0.0162
0.487 16 0.4868
48-40 13 0.0185
0.057 16 0.0567
18-49 15 0.0085
0.036 16 0.0359
18-50 11 0.0251
0.025 15 0.0058
The Figure 12 and Figure 13 show the best locations of shunt FACTS devices
to improve the FSS margin, i.e. with this location the system pass from
unstable situation to stable situation. Additionally, the reactive power (size) of
each location is calculated.
Figure 12. Best locations of shunt FACTS devices to improve the FSS margin under
contingencies on the line 22 and line 49
G07
07
23 05G04
04
G05
19
20
G06
06
22
68
21
65
62
63
G03
03
64
66
67
58
G02
02
60
59
57
56
52
37
27
24G09
29
0928
26 25
G08
08
54
G01
01
55
NETS
1
2
3
FACTS
FACTS
FACTS
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
15
42
G14
14
41
NYPSAREA#5
AR
EA#4
AREA# 3
FACTS
SM Reactive power [Mvar]
Bus SVC STATCOM SVC STATCOM0 0.5002 -
23 0.1414 0.1424 94.475 98.17021 0.1232 0.1259 94.754 100.34668 0.1169 0.1209 96.841 103.08319 0.0708 0.0737 87.877 90.25622 0.0317 0.0318 90.163 92.857
SM Reactive power [Mvar]
Bus SVC STATCOM SVC STATCOM0 0.0567 -
50 0.0316 0.0366 293.235 385.308
Faulted line 21-22 Outage line 40-48
29
Figure 13. Best locations of shunt FACTS devices to improve the FSS margin under
contingencies on the line 18 and line 86
On the other hand, for a fault on the line 48 there is not location suitable to
pass from unstable situation to stable situation on the system, seeFigure 14.
Figure 14. Fault on the line 41 on the system.
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
15
42
G14
14
41
NYPSAREA#5
AREA#4
AREA# 3
FACTS
SM Reactive power [Mvar]
Bus SVC STATCOM SVC STATCOM0 0.0359 -
50 0.0347 0.1117 198.104 213.106
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
15
42
G14
14
41
NYPSAREA#5
AR
EA
#4
AREA# 3
FACTS
SM Reactive power [Mvar]
Bus SVC STATCOM SVC STATCOM0 0.0251 -
32 0.6881 0.887 148.963 156.717
Faulted line 18-49 Outage line 18-50
G13
13
43
17
G12
12
36
61
30
53
47
4840
44
45
39
35
34
33
32
G11
11
31
38
51
50
G10
10
46
49
G16
16
18
G15
15
42
G14
14
41
NYPSAREA#5
AR
EA#4
AREA# 3
FACTS
?
Faulted line 40-41
30
3.5. Conclusions
The location and sizing for shunt FACTS devices is presented in order
to improve the first swing stability margin.
The location is determined with the Stability Margin and Instability
Margin.
In the most the cases the best allocation corresponds to busbars near to
the SDM.
The size is determined as the value of reactive power maximum in the
post-fault period.
The STATCOM is most effective in order to increase the first swing
stability.
31
Chapther 4
4. Optimal Location and Sizing of Shunt FACTS
Devices using Genetic Algorithm (GA)
4.1. Introduction
Genetic algorithms are search algorithms based on the mechanics of natural
selection and natural genetics. The GA differs from other methods because it
does not require auxiliary knowledge; it works with a coding of the parameter
set of the optimization problem. The increasing popularity of the GA can be
attributed to its simplicity, elegance, ease of implementation, and its proven
ability to often find good solutions for difficult high-dimensional function
optimization or combinatorial problems with continuous or discrete variables
[16].
The objective of the GA implemented here is to find the optimal location
and sizing of the shunt FACTS devices to maximize the objective function or
fitness function. The analysis for SVC and STATCOM devices is done
separately and theirs results compared afterwards. The steps involved in
implementation of the GA are described in this section.
4.2. Multi-Objective Formulation
In this Chapther, the optimal sizing and location of shunt FACTS devices are
found in order to maximize the first swing stability margin and voltage
stability margin or system loading, simultaneously using GA. The loading
margin and the transient stability margin are severely affected by faults on
lines of the power system. Therefore, some critical contingencies in the
system are analyzed. The multi-objective problem functions, the typical
constrains in a power system, and the critical contingencies are described in
this section.
32
4.2.1. Voltage Stability Margin or Loading Margin
Function
The loading margin, for a particular operating point, is the amount of
additional load, both active and reactive power, in a specific pattern of load
increase that would cause a voltage collapse [8]. Therefore, the loading
margin of a power system is an important measure of its proximity to voltage
collapse, and by increasing it; the voltage stability margin is also increased.
Thus, the first objective can be expressed as:
maxf1 = Loading(λ)
The system loading (λ) is calculated using the continuous power flow (CPF),
which increases the system load until the voltage collapse point is reached.
4.2.2. First Swing Stability Margin Function
The second objective function is the same shown in the Chapther 3and can be
expressed as:
max f2 = SM
or
max f2 = -IM
In other words, the f2 calculation depends on the stability of the system.
4.2.3. Problem Constrains
There are two kinds of constrains in a power system. The first one is the
equality constraints that represent the typical load flow equations and can be
expressed as:
g(x, u) = 0
33
The last one is the inequality constraints that represent the reactive power limit
of generators and the operating limits of the STATCOM and SVC, and can be
expressed as:
h(x, u) ≤ 0
The equality and inequality constraints must be satisfied during the
optimization procedure.
4.2.4. Critical Contingency
The critical contingencies analyzed are those that cause voltage instability or
angular instability in the power system. In this case, a contingency is
considered critical in voltage if the system loading under that contingency is 0.
On the other hand, a contingency is considered critical in angle if the angle of
at least one of the machines is higher than 180 or if the system has not
adequate damping in subsequent swings.
4.2.5. Multi-objective Optimization Problem
Considering the objectives, constraints and the critical contingencies, the
optimal sizing and location of the shunt FACTS devices can be formulated as
a nonlinear constrained multi-objective optimization problem as follows:
Subject to
g(x, u) = 0
h(x, u) ≤ 0
Where, ncc are the critical contingencies analyzed.
34
4.3. Genetic Algorithms and its Implementation
Genetic algorithms are search algorithms based on the mechanics of natural
selection and natural genetics. The GA differs from other methods because it
does not require auxiliary knowledge; it works with a coding of the parameter
set of the optimization problem. The increasing popularity of the GA can be
attributed to its simplicity, elegance, ease of implementation, and its proven
ability to often find good solutions for difficult high-dimensional function
optimization or combinatorial problems with continuous or discrete variables
[16].
The objective of the GA implemented here is to find the optimal location and
sizing of the shunt FACTS devices to maximize the objective function or
fitness function. The analysis for SVC and STATCOM devices is done
separately and theirs results compared afterwards. The steps involved in
implementation of the GA are described in this section.
4.3.1. Individual Configuration
The individual configuration is based on shunt FACTS device and it is
encoded in two parameters: location and rated value [9]. Suppose the FACTS
devices number to be analyzed is four, as shown in Figure 15.
nfacts=4
Location
Value
20 63 56 50
1.0 3.0 2.0 2.5
Figure 15. Individual configuration of shunt FACTS device.
Each location value is the number of the bus where the FACTS device is
located. In this case, the generation busbars are not considered. More than one
device cannot be placed in the same bus. The second value represents the rated
35
value of each shunt FACTS device in p.u.. Both location and rated value are
generated randomly.
4.3.2. Initial Population Configuration
The initial population configuration is generated by repeating the individual
configuration operation, nindividuals [9]. Suppose the population is formed by
four individuals, as shown in Figure 16.
28 34 68 45
3.0 2.0 3.0 2.5 30 42 67 60
1.5 2.0 3.0 3.0 35 42 57 52 0.5 1.0 2.0 1.5 20 63 56 50
1.0 3.0 2.0 2.5
nIndividuals=4
Figure 16. Initial population configuration of shunt FACTS device.
In order to obtain different solutions for each population, an individual must
not be repeated.
4.3.3. Fitness Function
The objective function or fitness function is evaluated for each individual of
the population, and is defined as follows:
Fitness = ftotal
The fitness function value obtained here is taken into account in applying
genetic operators.
36
4.3.4. Reproduction Operator
For each individual of the generation, a probability to be selected for the next
generation is calculated. Each probability depends on the fitness function
value. Individuals with greater fitness function will have a higher chance to be
selected or contribute one or more offspring into the next generation. To select
the individuals, independent random events between 0 and 1 are generated.
This process is called roulette wheel parent selection and may be viewed as a
roulette wheel where each individual of the population is represented by a
slice that is directly proportional to the individual’s fitness [9], [16], as shown
in Figure17.
P{Ind1}
P{Ind2}
P{Ind3}
P{Ind4}
f{Ind1}
f{Ind2}
f{Ind3}
f{Ind4}28 34 68 45
3.0 2.0 3.0 2.5 30 42 67 60
1.5 2.0 3.0 3.0 35 42 57 52 0.5 1.0 2.0 1.5 20 63 56 50
1.0 3.0 2.0 2.5
Figure17. Roulette wheel reproduction operator.
4.3.5. Crossover Operator
The individuals selected in the process mentioned above are called parents.
The crossover operator is applied to generate children from two parents. A
single point crossover is applied to generate children to the next generation
[9], [16], as seen in the schematic shown in Figure 18.
37
Parent1
30 42 …. 60 1.5 …. …. 3.0
28 34 68 45
3.0 2.0 …. 2.5
67 60 3.0 3.0
68 45
3.0 2.5 Child1
Child2Parent2
28 34 …. 45
3.0 2.0 …. 2.5 67 60 3.0 3.0
30 42 …. 60 1.5 2.0 …. 3.0
68 45 3.0 2.5
Before crossover After crossover
Crossover point
Figure 18. Single-point crossover point
The crossover point is generated randomly between 1 and system bus number.
The first child is formed with the first part of the parent1 and the second part
of the parent2. The second child is formed with the first part of the parent2
and the second part of the parent1. The parts of parents depend on crossover
point generated (see Figure 18).
4.3.6. Mutation operator
The mutation operator is applied to children generated in the crossover
process above. The mutation is introduced to change the location and size of
shunt FACTS devices, which are selected for independent random events [9].
The mutation point is generated randomly between 1 and the system bus
number, as shown in Figure 19.
Child3
Child430 42 …. 60 1.5 2.0 …. 3.0
68 45 3.0 2.5
28 34 …. 45
3.0 2.0 …. 2.5 67 60 3.0 3.0
30 42 …. 60 2.5 2.0 …. 3.0
68 29 3.0 2.5
28 37 …. 45
3.0 2.0 …. 2.5 67 60 3.0 1.0
Child1
Child2
Before mutation After mutation
Figure 19. Single-bit point mutation operator.
38
Single-bit point mutation is applied in this case, to generate the rest of the
children for the next generation [9], [16].
The GA implemented is summarized in the flow chart shown in Figure 20.
Evaluate the fitness
of each individual
Start
Input: nfacts
and nIndividuals
Generate initial
population
Stop criterion
reached?
Create new
generation using:
Reproduction,
Crossover,
Mutation
No
End
Print best
individual
Yes
Figure 20. Flowchart of GA implemented.
4.4. Testing Results
The proposed method is validated on a 68 bus test system, which is fully
explained in [17].
The shunt FACTS devices number (nfacts) is defined for each individual of a
generation, getting a random number between 1 and 8, i.e. the maximum
amount that can be allocated on this power system is set to 8 for each
individual. The shunt FACTS devices number to be allocated optimally in the
39
power system depends on the amount of lines improved during the
optimization process. A line is considered improved if a contingency on it
does not cause voltage instability or/and first swing instability.
The individual’s number of each generation (nindividual) is defined as four
and the stop criterion is defined by the generations number reached, which is
250.
4.4.1. Critical Contingencies Analyzed
The critical contingencies to be analyzed and that affect the two objective
functions are shown in Table VII. Line 18 and line 22 make the unstable
system in angle. The line 63 makes the unstable system in voltage and the rest
of the line affect both functions.
Table VII. Critical Contingencies for Voltage and Angle Stability
Line From Bus To Bus Function Affected
18 18 49 f2
22 21 22 f2
48 40 41 f1 and f2
49 40 48 f1 and f2
63 45 51 f1
86 18 50 f1 and f2
4.4.2. GA Solution
In Table VIII, the optimal location and sizing for each device is shown. In the
SVC case, four units in the whole system are necessary to improve most of the
lines and to maximize the fitness function. A smaller STATCOM devices
number (three) is needed in comparison to SVC devices. The STATCOMs
total sizing is smaller than SVCs.
40
Table VIII. Optimal Location and Sizing for Each Shunt Facts Device
Device Location (Bus) Size (p.u.)
18 3.5
SVC 21 1.5
32 1.5
50 4.5
21 1.5
STATCOM 32 2.5
50 4.5
Table VIIIshows the lines improved with the GA solution for both devices.
Whit this solution, four lines are improved for angle stability and two lines are
improved for voltage stability.
Table IX. Improved Contingencies for Both Devices
Line Function Affected Function Improved
18 f2 f2
22 f2 f2
48 f1 and f2 f1
49 f1 and f2 f1 and f2
63 f1 -
86 f1 and f2 f2
The results show that not all the lines can be improved, e.g. under a fault on
line 48, the system is stable in voltage, but unstable in the first swing. If a fault
occurs on line 86, the system is stable in the first swing but unstable in
voltage, i.e. the use of shunt FACTS devices in these cases is not suitable.
Additionally, line 63 cannot be improved, i.e. for any configuration of shunt
FACTS devices a fault in this line, still makes the unstable system.
Table Xshows the objective functions and fitness function value for each
shunt FACTS device obtained during the optimization process. Despite the
SVC devices number found in the solution is greater than the STATCOM
controller’s number; the difference shown between the fitness function values
is small.
41
Table X. Objective and Fitness Function Value for Each Device
Device f2 f2
Fitness
SVC 0.7812 2.1436 2.9248
STATCOM 0.7984 1.9841 2.7785
Figure21shows the behavior of the fitness functions for both devices during
the optimization process.
Figure21. Fitness functions during optimization process.
4.4.3. Application of the GA Solution
In order to check the final solution, it is implemented on the test system and
the results are showed below. Figure 22 and Figure 23 show the loading when
the SVCs and STATCOMs are installed on the system, respectively. The base
case corresponds a system loading when it does not present line outages. The
system loading is higher in the base case; in two cases, outage of the line 48
and line 49, the system loading is greater than zero, therefore these are
improved with the optimal location and sizing of the shunt FACTS devices.
0 50 100 150 200 2500.5
1
1.5
2
2.5
3
Generation
Fit
ness
STATCOM
SVC
42
With the remainder of the lines analyzed, the system presents a loading of zero
for both devices.
Figure 22.System loading with SVC.
Figure 23.System loading with STATCOM.
0 0.2 0.4 0.6 0.8 1 1.2 1.40.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Loading[]
Vo
lta
ge[p
.u.]
Base Case
Outage Line 48
Outage Line 49
0 0.2 0.4 0.6 0.8 1 1.2 1.40.7
0.8
0.9
1
1.1
1.2
Loading[]
Volt
age[p
.u.]
Base Case
Outage Line 48
Outage Line 49
43
Figure 24 shows the angles of the severely disturbed machines (SDMs)
during different faults on the system, e.g. for a fault on line 22, the seventh
machine behaves as the SDM. These angles reach the 180 during the
simulation time, therefore the system is unstable. For faults on lines 18 and 86
the angles reached the 180° in a longer simulation time.
Figure 24. Machine angles during line faults without shunt FACTS devices.
Figure 25 and Figure 26 show the machine angles during faults on the critical
contingencies with the SVC and STATCOM allocated and sized (see Table II)
on the system, respectively. All the lines that affected the system in angle
stability without the presence shunt FACTS devices are improved except line
48.
5 10 15 20 2560
80
100
120
140
160
180
Time [s]
An
gle
[°]
Fault Line 18
Fault Line 22
Fault Line 48
Fault Line 49
Fault Line 86
G11
G16
G7
G16
G16
44
Figure 25. Machine angles during line faults with SVC.
Figure 26. Machine angles during line faults with STATCOM.
5 10 15 20 25
60
80
100
120
140
160
180
Time [s]
An
gle
[°]
Fault Line 18
Fault Line 22
Fault Line 48
Fault Line 49
Fault Line 86
5 10 15 20 25
60
80
100
120
140
160
180
Time [s]
An
gle
[°]
Fault Line 18
Fault Line 22
Fault Line 48
Fault Line 49
Fault Line 86
45
4.5. Conclusions
In this Chapther a new method to find the optimal location and sizing of
the shunt FACTS devices simultaneously using GA, has been proposed.
The method optimizes a multi-objective problem which objective
functions are: loading margin and first swing stability margin. These objective
functions are solved for some critical contingencies in the system.
It is important to note that in certain line faults in the system, the shunt
compensation does not result in optimal reactive power reinforcement for
voltage stability enhancement or first swing stability. However, the proposed
method finds the more suitable size and location in order to improve most of
the lines in the system and to maximize the objective function.
Between the two shunt FACTS devices used, the STATCOM is more
effective than SVC if the comparison is made on the required performances
and not just the size or the cost.
The proposed method has been applied on a 68 bus test system.
46
Chapther 5
5. General Conclusions
In this thesis a new method to find the optimal location and sizing of the
shunt FACTS devices simultaneously using GA, has been proposed.
The location and sizing of shunt FACTS devices to improve the voltage
stability margin or system loading is determined.
The location and sizing of shunt FACTS devices to improve first swing
stability margin is determined.
The location and sizing when an objective is analyzed itself are different
when the multi-objective problem is solved.
For certain line faults in the system, the shunt compensation does not
result in optimal reactive power reinforcement for voltage stability
enhancement or first swing stability.
In cases where the solution was not effective, use of other-device in the
system is recommended.
47
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