Optimal Compensation of Reactive Power in Transmission Networks using PSO, Cultural and Firefly Algorithms
Subhash Shankar Zope1and 2Dr. R.P. Singh
1Sri Satya Sai University of
Technology & Medical Sciences. Opposite OILFED, Bhopal Indore Highway,
Pachama, Sehore (India) 466001
2 Sri Satya Sai University of
Technology & Medical Sciences.
Opposite OILFED, Bhopal Indore Highway,
Pachama, Sehore (India) 466001
Abstract
Increased demand for electrical energy and free
market economies for electricity exchange, have
pushed power suppliers to pay a great attention to
quality and cost of the latter, especially in
transmission networks. To reduce power losses
due to the high level of the reactive currents
transit and improve the voltage profile in
transmission systems, shunt capacitor banks are
widely used. The problem to be solved is to find
the capacitors optimal number, sizes and locations
so that they maximize the cost reduction. This
paper is constructed as a function of active and
reactive power loss reduction as well as the
capacitor costs. To solve this constrained non-
linear problem, a heuristic technique, based on
the sensitivity factors of the system power losses,
has been proposed. The proposed algorithm has
been applied to numerous feeders and the results
are compared with those of the authors having
International Journal of Pure and Applied MathematicsVolume 114 No. 9 2017, 367-388ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu
367
treated the problem. The optimal location SVC is
studied on the basis of Heuristic methods; Particle
Swarm Optimization (PSO), Cultural Algorithm
(CA), and Firefly Algorithm (FF) to minimize
network losses. Validation of the proposed
implementation is done on the IEEE-14 and
IEEE-30 bus systems.
Key Words andPhrases:CA, FDL, FF, PSO,
SVC.
1 Introduction
The rapid growth of the population as well as the industry are
primarily the factors influencing the consumption of electrical
energy which is on the other hand continuously increasing.
Since storing the electrical energy is a challenging task, it
requires a permanent balance between consumption and
production for that it is at first sight necessary to increase the
number of power stations, and of the various structures
(Transformers, Transmission lines, etc.), this leads to an
increase in cost and a degradation of the natural environment
[1].
1.1 Classification of the Variables of Power Flow Equations
In an electrical network each busbar is connected with four
fundamental magnitudes: The modulus of the voltage , the
phase of the voltage , the active power injected and the
reactive power injected . It is very important to note that for
each busbar; two variables must be specified beforehand and the
other two must be calculated. These different variables each
have a name following their role in the electrical network. The
state of the system is determined only after calculating the
values of the state variables [2]:
The disturbance variables: These are uncontrolled variables
representing the power demands of loads, the perturbation
variables are: and | |.
The state variables: Modules and phases are called unknown
state variables which characterize the state of the system, these
variables are: | |and .
Control variables: In a generic way, the active and reactive
powers injected are called control. It is also possible, depending
on the case, to consider tensions at the generation nodes or the
transformation ratios of transformers with load adjuster as
control variables, these variables are: [2].
1.3 Types of busbars
In the power flow analysis, the busbars in the system are
classified in three categories:
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Reference busbar (slack bus): It is also called the swing bus or
swing bus, it is a fictional element created for the study of the
distribution of power, and it has for role to provide the
additional power necessary to compensate for transmission
losses, as these are not known in advance. In general, by
convention, this busbar is identified by the set of busbars𝑁° = 1
connected to a voltage source from which the module | | and
phase (δ) of the voltage are known, these values are taken as
references = 1𝑝𝑢 and δ = 0°. The active powers ( ) and ( ) are
therefore unknown and must be calculated after solving the
problem of the power flow [2] and [3].
Busbars (control bus): Also known as generator or voltage
controlled busbars; they may include games of barriers to which
generators, capacitor banks, static compensators or
transformers with adjustable plug are connected to control the
voltage. The parameters specified here are: the active power ( )
and the voltage modulus ( ) from which the term: busbars therefore the remaining parameters must be calculated ( ) and
(δ) [2] and [3].
Loadbars (Loadbus): Also called the busbars (P Q), the specified
values are the active powers ( ) and ( ), the values to be
computed are the modulus and the phase (δ) of the voltage [2]
and [3].
This paper is organized into nine sections. Section 2 gives
analytical methods. Then in 3, relative unit (Static VAr
Compensator) is explained. Section 4 details about modeling
branches and loads. Section 5 explains the optimization of the
reactive energy compensation by fixed batteries using heuristic
method. Section 6 describes load flow analysis using Newton-
Raphson method. Section 7 provides optimal location using
heuristic techniques. Results and analysis is described in
section 8 and finally, a conclusion summarizes the contributions
of the paper.
2 Analytical Methods
The pioneer in the field is Cook [4]. In 1959, he studied the
effects of capacitors on power losses in a radial distribution
network where the charges are uniformly distributed. It
considered the reduction of power losses as an objective function
by considering a periodic reactive charging cycle. Cook then
developed a network of convenient curves to determine the most
economical power of the capacitor bank and the location of the
capacitor bank on the line. The equation giving the optimal
location to be assigned to a specified size battery is given by:
2.1 Methods of Digital Programs
The development of methods has led researchers to become
increasingly interested in the optimization of reactive energy
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compensation. Therefore, they have developed numerical
methods for the analysis of the electrical network.
Baran and Wu [5] in 1989, presented a method for solving the
problem of placing capacitor banks in distribution networks. In
this problem, the locations of the batteries, their sizes, their
types, the stresses of the voltage and the variations of the load
are taken into account. The problem is considered to be a non-
linear programming problem where load flow is explicitly
represented.
To solve this problem, it is broken down into a slave problem
and a master problem. They exploit for this purpose the
property of optimization where:
𝑢 { 𝑢 } (1)
3 Relative Unit (Static
VArCompensator)
The normalization of the resistance of the line is obtained by
relating it to a calculated basic resistance by means of the
voltage and the power . If the base voltage is given
in kV and the power in kVA then, this resistance is given by:
(2)
The normalized resistance is then:
(3)
Standardized load ratings are obtained by:
{
(4)
4 Modeling Branches and Loads
4.1 Modeling Branches
The distribution networks have a radial configuration and
consist of a set of branches. Each branch of this network is
modeled as a series resistor with a pure inductance. The
impedance of any branch "i" of this network (see Fig. 1) is given
by:
(5)
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Fig. 1. Single line diagram of a branch
The shunt admittances are negligible because the line is of
medium voltage.
4.2 Modeling of Loads
The loads are generally modeled as being voltage dependent. We
write for the active and reactive powers of a charge placed at the
node "i" the following expressions:
(
)
(6)
(
)
(7)
Where,
and are the nominal active and reactive powers.
is the nominal voltage.
and are the active and reactive power of the load at node ‘ ’ for a voltage equal to .
The coefficients and determine the character of the load.
If the coefficients and are both zero, the load is considered to
be constant power. If, on the other hand, and are equal to 1,
the load is considered to be constant current. When they are
equal to 2 the load is considered to have a constant impedance.
In the remainder, and will be zero, i.e. consider constant
power loads.
The apparent power of the load connected to the node is in this
case:
(8)
5 Method of Solution
The voltage drop method is an iterative method. Its principle
consists in calculating, first and for each section of the line, the
powers at the end of the branch, the losses of active and reactive
powers and the powers at the beginning of the branch. From
these, the currents of the branches are determined by raising
the line to the source. These currents are calculated from the
estimated values of the voltages, the powers at the beginning of
i
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the branch and the values of the impedances of each line section
between two successive busbars.
5.1 Formulation of the Problem
Power losses, low power factor and degradation of the voltage
profile are the result of strong current flow in power
systems.These phenomena are more pronounced in distribution
networks where the branch currents are stronger compared to
those circulating in the transport networks. The problem is
therefore to decide the number of batteries, their powers and
their locations which would make an objective function "F"
maximum. This objective therefore makes the problem of
reactive energy compensation an optimization problem.
However, owed to the discrete nature of the battery sizes and
their locations, this problem is non-linear with constraints. It is
generally modelled as follows:
{
𝑢 𝑢
𝑢 𝑢 𝑢
(9)
Where,
: is the objective function to maximize.
: is the equality constraint. It is the set of equations of the
power flow
: is the control variable vector
𝑢: is the state variable vector.
5.2 Objective Function
The objective function on which all the authors who have dealt
with the problem of optimization of reactive energy
compensation is the so-called "economic return" function or cost
reduction noted as "∆S". Mathematical expression is given by:
∑ (10)
Where,
: is the total number of batteries installed.
: is the cost of kW produced (₹ /kW).
: is the annual price of kVAr installed depreciation and life
included.
: is the size of the installed battery at node " ".
: is the reduction of the active power losses.
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5.3 Reduction of Active Power Losses
The reduction of the power losses due to a battery "k" is equal to
the difference of the losses of active power in the network before
and after the installation of the said capacitor bank. It is given
by:
(11)
Where,
: are the active power losses in line before compensation.
: are active power losses in line after compensation.
5.4 Reduction of Reactive Power Losses
The reduction of the reactive power losses due to a battery
installed at node " " of the distribution line is defined by the
difference between the losses before and after the installation of
batteries in question of capacitors. It is given by:
(12)
Where,
: are the losses of reactive power in line before
compensation.
: are the losses of reactive power in line after compensation.
5.5 Reactive Power Losses
The losses of reactive power in a distribution network line
composed of n branches are given by the following formula:
∑
(13)
Where,
is the reactance of branch
is the line current of the branch.
As with the active power losses, the active and reactive
components of the branch current thus allow to write the losses
of reactive power as follows:
∑
∑
(14)
The losses of reactive power when a capacitor bank is placed on
a node are given by:
∑
∑
∑
(15)
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The reduction of reactive power losses by calculating the
difference between equation (14) and equation (15), will be equal
to:
∑
∑ (16)
5.6 Heuristic Method
Heuristic methods are based on experience and practice. They
are easy to understand and simple in their implementation.
They use sensitivity factors which they incorporate into
optimization methods in order to achieve qualitative solutions
with small computational efforts. Since the problem of
determining the suitable battery locations has been separated
from that of optimum power determination since the locations
are determined by the sensitivity factors then the size
calculation is generally modeled as follows:
{
𝑢
(17)
5.7 New Modeling of the Problem
By substituting the constraint on the tension with that made on
the branch current, the new mathematical model of the problem
becomes:
{
𝑢
(18)
5.8 Optimal Operation of Batteries
The reduction in power losses for a given node "k" is defined as
the difference between the power losses before canceling the
reactive current of the load at node " " and after the latter has
been canceled. It is given by:
(19)
The power losses before the cancellation of the reactive current
of the load at the node " " are given by:
∑
∑
(20)
The power losses after the cancellation of the reactive current of
the load at the node " " are given by:
∑
∑
∑
(21)
After simplification, the reduction in power losses will have the
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following expression:
∑
∑
(22)
5.9 Determination of Optimal Sizes
To calculate the optimum sizes of the batteries, the currents
they generate are first determined. This current is calculated so
as to make the objective function the maximum cost
reduction. This current is acquired by undertaking the
accompanying condition:
(23)
The expression of the current is then given by:
∑
∑
∑
∑
(24)
The initial optimum power is calculated by the following
expression:
(25)
The maximum value of the cost reduction in this case:
* ∑ ∑
+
( ∑ ∑
)
(26)
The value of the equivalent power loss reduction is given by:
(∑
)
(∑
)
( ∑ ∑
)
∑ [ ∑
∑
]
( ∑ ∑
)
(27)
5.10 Solution Strategy
By optimally compensating the reactive energy we expect the
battery locations to be busbars in the network and that the
optimum battery power is available commercially or multiple of
these batteries. If the constraint on the locations, which can
only be busbars of the network, has found a solution by means
of sensitivity factors, the optimum powers of the batteries
remain to be determined by solving the problem with the
following constraints:
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{
𝑢
(28)
The detailed solution algorithm of the problem of determination
will be detailed in the following section.
5.11 Calculation Algorithm
The algorithm for solving the overall problem, i.e., the suitable
locations of the capacitor banks and their sizes is detailed in
what follows. A MATLAB 14a environment program has been
developed for this purpose.
Step 1: Read the network data.
Step 2: Perform the program of the power flow before
compensation to determine the active and reactive power losses,
branch currents, node voltages and their phases at the origin.
Step 3: Initialize reduction of power losses and cost.
Step 4: as long as the reductions in power losses and cost are
positive.
Step 4.1: Determine the sensitivities of the nodes according to
equation (22) and rank them in descending order.
Step 4.2: If the most sensitive node considered has already
received a battery, ignore it.
Step 4.3: Calculate the initial value of the optimum size of the
battery to be placed there, the reduction of the cost and the
reduction of the power losses.
Step 4.4: Perform load flow to update electrical quantities
(voltage, current, power).
Step 4.5: Adjust the optimal size of the battery.
Step 4.6: If the battery size is negative, smaller than the
smallest standard battery or greater than the total power and
the reduction of negative power losses then:
Step 4.6.1: Remove the battery.
Step 4.6.2: Give the voltages at their origin and branch currents
the values d before the battery.
Step 4.7: Otherwise, take as the optimum size of the battery, the
lower standard size where higher giving the greatest cost
reduction.
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Step 4.8: Re-calculate the load flow.
Step 4.9: If the battery produces overcompensation then:
Step 4.9.1: Replace the standard battery with a smaller one that
does not overcompensate.
Step 4.9.2: Test if the battery is not smaller than the smallest
standard battery.
Step 4.9.3: Perform load flow and calculate the reduction in
power losses and cost based on the actual installed kVAr power.
Step 4.10.4: Verify that the reduction in power loss and cost
reduction are positive.
Step 4.10: End if
Step 4.11: Go to Step 4
Step 5: Display the results.
6 Load Flow Analysis using Newton-
Raphson Method
This method requires more time per iteration where it does not
requires only a few iterations even for large networks. However,
it requires storage as well as significant computing power. Let
us assume:
(29)
∑
(30)
(31)
We know that:
Equation (29) then becomes:
∑
(32)
By separating the real and the imaginary part, one obtains:
{ ∑
∑
(33)
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Positions:
(34)
(35)
Where,
{
Then, the equation (33) becomes:
{ ∑
∑
(36)
Where,
{ ∑
∑
(37)
It is a system of nonlinear equations. The active power and
the reactive power are, and the real and imaginary
components of the voltage and are unknown for all Bus bars
except the reference bus bar, where the voltage is specified and
fixed. The Newton-Raphson method requires that non-linear
equations be formed of expressions linking the powers and the
components of the voltage.
[
]
[
|
|
|
]
[
]
(38)
Where the last set of bars is the reference bar. The outline of the
matrix is given by:
*
+ *
|
+ *
+ (39)
Or
[
] [
] (40)
Where [J] is the Jacobian of the matrix. are the
differences between the planned values and the values
calculated respectively for active and reactive powers.
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Equation (37) can be written as follows:
{
∑
∑
(38)
From where, one can draw the elements of the Jacobian:
The diagonal elements of : ∑
The non-diagonal elements of
The diagonal elements of ∑
The non-diagonal elements of
The diagonal elements of ∑
The non-diagonal elements of
The diagonal elements of ∑
The non-diagonal elements of
(39)
(40)
Because of the quadratic convergence of the Newton-Raphson
method, a solution of accuracy can be achieved in just a few
iterations. These characteristics make the success of the Fast
Decoupled Load Flow and the Newton-Raphson.
7 Optimal Location using Heuristic
Techniques
In the proposed system, the location of SVC in a particular bus
system is decided by PSO, CA, FF algorithms. The objective
function is minimized using the abovementioned techniques.
7.1 Particle Swarm Optimization (PSO)
James Kennedy and Russell C. Eberhart proposed a PSO
approach in 1995. This approach is a heuristic method [6]. The
evaluation of candidate solution of current search space is done
on the basis of iteration process (as shown in Fig. 2). The
minima and maxima of objective function is determined by the
candidate’s solution as it fits the task’s requirements. Since PSO
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algorithm do not accept the objective function data as its inputs,
therefore the solution is randomly away from minimum and
maximum (locally/ globally) and also unknown to the user. The
speed and position of candidate’s solution is maintained and at
each level, fitness value is also updated. The best value of
fitness is recorded by PSO for an individual record. The other
individuals reaching this value are taken as the individual best
position and solution for given problem. The individuals
reaching this value are known as global best candidate solution
with global best position. The up gradation of global and
individual best fitness value is carried out and if there is a
requirement then global and local best fitness values are even
replaced. For PSO’s optimization capability, the updation of
speed and position is necessary. Each particle’s velocity is
simplified with the help of subsequent formula:
(41)
Fig. 2. Flow chart of PSO algorithm [6]
7.2 Cultural Algorithm
Cultural algorithm corresponds to modeling inspired by the
evolution of human culture [7]. Thus, just as we speak of
biological evolution as the result of a selection based on genetic
variability, we can speak of a cultural evolution resulting from a
selection exercising on the variability Cultural development.
From this idea, Reynolds developed a model whose cultural
evolution is considered as a process of transmission of
experience at two levels: a micro-evolutionary level in terms of
transmission of genetic material between individuals of a
population and a macro level -evolutionary in terms of
No
Start
Initialization on early searching points of all agents
Assessment of searching points of all agents
Amendment of every searching point using state equation
Extent to extreme iteration
Stop
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knowledge acquired on the basis of individual experiences. The
following figure presents the basic CA framework.
As Fig. 3 shows, the population space and the belief space can
evolve respectively. The population space consists of the
autonomous solution agents and the belief space is considered
as a global knowledge repository. The evolutionary knowledge
that stored in belief space can affect the agents in population
space through influence function and the knowledge extracted
from population space can be passed to belief space by the
acceptance function.
Fig. 3. CA framework [8]
7.3 Firefly Algorithm
Fireflies are small flying beetles capable of producing a cold
flashing light for mutual attraction. In the common language
between fireflies, they are also used synonymous lighting bugs
or glow worms. These are two beetles that can emit light, but
fireflies are recognized as species that have the ability to fly.
These insects are able to produce light inside their bodies
through special organs located very close to the surface of the
skin. This light production is due to a type of chemical reaction
called bioluminescence [9].
Principle of operation of the algorithm of Fireflies
The algorithm takes into account the following three points:
• All fireflies are unisex, which makes the attraction
between these is not based on their gender.
• The attraction is proportional to their brightness, so for
two fireflies, the less bright will move towards the
brighter. If no firefly is luminous that a particular
Belief Space
Update
Acceptance Function Influence Function Communication
Protocol
Population Space
Inherit
Evaluation Adaption,
Reproduction
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firefly, the latter will move randomly.
• The luminosity of the fireflies is determined according
to an objective function (to be optimized).
Based on these three rules, the Firefly algorithm is as follows:
Fig. 4. Pseudo code for Firefly Algorithm
8 Results Analysis
On the IEEE-14 and IEEE-30 bus test systems (shown in Fig. 5
and Fig. 6) the proposed heuristic techniques (PSO, CA and FF)
have been tested.
Fig. 5. Single line diagram of the IEEE-14 bus test system
G
C
G
C
C
1
2
3
4
5
6 7 8
9
10 11
12
1
3 1
4
G Gener
ators C Synchro
nous
Compe
nsators
Three Winding
Transformer
Equivalent
C
4 8
9
7
Define an objective function Generate a population of fireflies Define the intensity of light at a point by the objective function Determine the absorption coefficient As long as ( Max Generation) For to For to If
Move the firefly to the firefly End if Vary the attraction as a function of the distance via 𝑝 Evaluation of new solutions and updating light intensity End For End For Classify fireflies and find the best solution End as long as View results
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Fig. 6. Single line diagram of the IEEE-30 bus test system
Fig. 7. Active & Reactive power losses in IEEE-14 bus system
using PSO
NRPF NRPF with SVC (PSO)0
10
20
30
40
50
60
Pow
er
Loss (
MW
& M
Var)
Active & Reactive Power Losses in IEEE Bus System
13.7214
56.5404
13.5531
54.7546P
Q
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Fig. 8. Active & Reactive power losses in IEEE-30 bus system
using PSO
Fig. 9. Active & Reactive power losses in IEEE-14 bus system
using CA
NRPF NRPF with SVC (PSO)0
10
20
30
40
50
60
70
Pow
er
Loss (
MW
& M
Var)
Active & Reactive Power Losses in IEEE Bus System
17.8162
69.4087
17.8162
69.4087
P
Q
NRPF NRPF with SVC (CA)0
10
20
30
40
50
60
Pow
er
Loss (
MW
& M
Var)
Active & Reactive Power Losses in IEEE Bus System
13.7214
56.5404
12.789
53.1876P
Q
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Fig. 10. Active & Reactive power losses in IEEE-30 bus system
using CA
Fig. 11. Active & Reactive power losses in IEEE-14 bus system
using FF
NRPF NRPF with SVC (CA)0
10
20
30
40
50
60
70
Pow
er
Loss (
MW
& M
Var)
Active & Reactive Power Losses in IEEE Bus System
17.8162
69.4087
16.764
67.678
P
Q
NRPF NRPF with SVC (FF)0
10
20
30
40
50
60
Pow
er
Loss (
MW
& M
Var)
Active & Reactive Power Losses in IEEE Bus System
13.7214
56.5404
13.789
54.1876P
Q
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Fig. 12. Active & Reactive power losses in IEEE-30 bus system
using FF
Table 1. Comparative analysis for IEEE-14 bus system
Heuristic
Method
Active
Power
Loss
Reactive
Power
Loss
PSO 13.5531 54.7546
Cultural
Algorithm
12.789 53.1876
Firefly
Algorithm
13.789 54.1846
Table 2. Comparative analysis for IEEE-30 bus system
Heuristic
Method
Active
Power
Loss
Reactive
Power
Loss
PSO 17.8162 69.4087
Cultural
Algorithm
16.764 67.678
Firefly
Algorithm
17.162 69.076
9 Conclusion
In our work in this paper, we presented a solution for the
problem of the circulation of strong reactive currents in
balanced distribution networks. A heuristic solution technique
based on a loss-of-power sensitivity factor has been proposed. In
this method, the choice of the candidate nodes to receive the
capacitor banks is arbitrated by the sensitivity of the power
losses of the entire electrical system studied to the reactive load
NRPF NRPF with SVC (FF)0
10
20
30
40
50
60
70
Pow
er
Loss (
MW
& M
Var)
Active & Reactive Power Losses in IEEE Bus System
17.8162
69.4087
17.162
69.076
P
Q
International Journal of Pure and Applied Mathematics Special Issue
386
current of each node. The most sensitive node is therefore the
one whose reactive current of charge produces the most loss
reduction.
During this work, the problem of the power flow in the
distribution networks, which is a prerequisite for the conduct of
the reactive energy compensation, is also taken care of, the
calculation of the power flow is imperative. An iterative method
has been developed for this purpose where a technique specific
to us has been given to recognize the configuration of the
network. Load flow analysis is also done using Newton-Raphson
method. Three Heuristic methods are used to optimize the
location of SVC using the MATLAB model; Particle Swarm
Optimization, Cultural Algorithm and Firefly algorithm. The
tests were performed taking SVC as the FACTS device. It was
found that the Cultural Algorithm has less power losses as
compared to other methods.
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power dispatch considering FACTS devices. Electric Power
Systems Research, 46(3), pp.251-257.
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by load cycle. Transactions of the American Institute of
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