International Electrical Engineering Journal (IEEJ) Vol. 4 (2013) No. 2, pp. 1049-1058 ISSN 2078-2365 http://www.ieejournal.com/
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
A Modified Shuffled Frog-Leaping
Optimization
Algorithm for Solving Optimal Reactive
Power Dispatch Problem
K. Lenin , B.Ravindranath Reddy ,M.Surya Kalavathi
Electrical and Electronics Engineering, Jawaharlal Nehru technological university Kukatpally,
Hyderabad 500 085, India.
Email : [email protected]
Abstract-: This paper presents an algorithm for solving
the multi-objective reactive power dispatch problem in
a power system. Modal analysis of the system is used
for static voltage stability assessment. Loss
minimization and maximization of voltage stability
margin are taken as the objectives. Generator terminal voltages, reactive power generation of the capacitor
banks and tap changing transformer setting are taken
as the optimization variables. This paper introduces a
new search-acceleration parameter into the
formulation of the original shuffled frog leaping (SFL)
algorithm to create a modified form of the shuffled
frog algorithm (MSFL) for solving above problem. .
The shuffled frog-leaping algorithm draws its
formulation from two other search techniques: the
local search of the ‘particle swarm optimization’
technique; and the competitiveness mixing of
information of the ‘shuffled complex evolution’
technique.
Key words —Optimal reactive power, Transmission loss,
Evolutionary algorithms, Shuffled frog leaping, shuffled
complex evolution, Particle swarm, Optimization.
1.0 INTRODUCTION
Optimal reactive power dispatch problem is one of the
difficult optimization problems in power systems. The
sources of the reactive power are the generators,
synchronous condensers, capacitors, static compensators
and tap changing transformers. The problem that has to be
solved in a reactive power optimization is to determine the
required reactive generation at various locations so as to
optimize the objective function. Here the reactive power
dispatch problem involves best utilization of the existing
generator bus voltage magnitudes, transformer tap setting
and the output of reactive power sources so as to minimize
the loss and to enhance the voltage stability of the system.
It involves a non linear optimization problem. Various
mathematical techniques have been adopted to solve this
optimal reactive power dispatch problem. These include
the gradient method [1-2], Newton method [3] and linear
programming [4-7].The gradient and Newton methods
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
suffer from the difficulty in handling inequality
constraints. To apply linear programming, the input-
output function is to be expressed as a set of linear
functions which may lead to loss of accuracy. Recently
global Optimization techniques such as genetic algorithms
have been proposed to solve the reactive power flow problem [8, 9].
The optimization of systems and processes is very
important to the efficiency and economics of many science
and engineering domains. Optimization problems are
solved by using rigorous or approximate mathematical
search techniques. Rigorous approaches have employed
linear programming, integer programming, dynamic
programming or branch-and-bound techniques to arrive at
the optimum solution for moderate-size problems.
However, optimizing real-life problems of the scale often
encountered in engineering practice is much more
challenging because of the huge and complex solution
space. Finding exact solutions to these problems turn out
to be NP-hard. This kind of complex problem requires an
exponential amount of computing power and time, as the
number of decision variables increases (Lovbjerg 2002). To overcome these problems, researchers have proposed
approximate evolutionary-based algorithms as a means to
search for near-optimum solutions. Evolutionary
algorithms (EAs) are stochastic search methods that mimic
the metaphor of natural biological evolution and/or the
social behavior of species. The optimized behavior of such
species is guided by learning, adaptation and evolution
(Lovbjerg 2002). Researchers have developed
computational systems that mimic the efficient behavior of
species such as ants, bees, birds and frogs, as a means to
seek faster and more robust solutions to complex
optimization problems. The first evolutionary based
technique introduced in the literature was the genetic
algorithm (Holland 1975). In an attempt to reduce
processing time and improve the quality of solutions,
particularly to avoid local optima, various genetic
algorithm improvements and other EAs have been
proposed during the past 10 years, with the latest and perhaps most promising technique being the shuffled frog-
leaping (SFL) algorithm (Eusuff and Lansey 2003,
Elbeltagi et al. 2005). This paper first presents the SFL
algorithm. Then, a new search – acceleration parameter is
added to the original formulation to create a modified
shuffled frog-leaping (MSFL) algorithm.In this paper, we
propose this powerful algorithm for solving reactive
power dispatch problem .The effectiveness of the
proposed approach is demonstrated through IEEE-30 bus
system. The test results show the proposed algorithm gives
better results with less computational burden and is fairly
consistent in reaching the near optimal solution
In recent years, the problem of voltage stability and
voltage collapse has become a major concern in power
system planning and operation. To enhance the voltage
stability, voltage magnitudes alone will not be a reliable
indicator of how far an operating point is from the collapse
point [11]. The reactive power support and voltage
problems are intrinsically related. Hence, this paper
formulates the reactive power dispatch as a multi-
objective optimization problem with loss minimization
and maximization of static voltage stability margin
(SVSM) as the objectives. Voltage stability evaluation
using modal analysis [12] is used as the indicator of
voltage stability.
2.0 VOLTAGE STABILITY EVALUATION
2.1 Modal analysis for voltage stability evaluation
Modal analysis is one of the methods for voltage stability
enhancement in power systems. In this method, voltage
stability analysis is done by computing eigen values and
right and left eigen vectors of a jacobian matrix. It
identifies the critical areas of voltage stability and
provides information about the best actions to be taken for
the improvement of system stability enhancements. The
linearized steady state system power flow equations are
given by.
[
] [
] (1)
where
ΔP = Incremental change in bus real power.
International Electrical Engineering Journal (IEEJ) Vol. 4 (2013) No. 2, pp. 1049-1058 ISSN 2078-2365 http://www.ieejournal.com/
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
ΔQ = Incremental change in bus reactive
Power injection Δθ = incremental change in bus voltage angle.
ΔV = Incremental change in bus voltage
Magnitude
Jpθ , J PV , J Qθ , J QV jacobian matrix are the sub-matrixes
of the System voltage stability is affected by both P
and Q. However at each operating point we keep P
constant and evaluate voltage stability by considering
incremental relationship between Q and V.
To reduce (1), let ΔP = 0 , then.
ΔQ=[JQV − JQθ J Pθ -1
J PV ]ΔV = J R ΔV (2)
ΔV = J -1
ΔQ (3)
Where
J R= (J QV − J Qθ J Pθ-1
J PV) (4)
J R is called the reduced Jacobian matrix of the system.
2.2 Modes of Voltage instability:
Voltage Stability characteristics of the system can be
identified by computing the eigen values and eigen
vectors
Let
J R = ξ ∧ η (5)
Where,
ξ = right eigenvector matrix of JR
η = left eigenvector matrix of JR
∧ = diagonal eigenvalue matrix of JR and
JR−1
=ξ∧−1η
(6)
From (3) and (6), we have
ΔV=ξ∧−1ηΔQ
(7)
Or
ΔV = ∑i ξi ηi ΔQ (8)
λi
where ξi is the ith column right eigenvector and η the ith
row left eigenvector of JR.
λi
is the ith eigen value of JR.
The ith
modal reactive power variation is,
ΔQmi=Kiξi (9)
where,
K i = ∑j
ξ ij2
– 1 (10)
Where
ξji is the jth element of ξi
The corr esponding i t h modal vol ta ge variat ion
i s
ΔVm i= [1/ λ i ] ΔQ m i (11)
It i s seen that , when the react ive po wer
variat ion i s along the direction of ξi the
corresponding voltage variation is also along the same
direction and magnitude is amplified by a factor which
is equal to the magnitude of the inverse of the ith
eigenvalue. In t h i s s e n s e , t h e magnitude of each
eigenvalue λi determines the weakness of the
corresponding modal voltage. The smaller the
magnitude of λi, the weaker will be the
corresponding modal voltage. If | λi | =0 the ith
International Electrical Engineering Journal (IEEJ) Vol. 4 (2013) No. 2, pp. 1049-1058 ISSN 2078-2365 http://www.ieejournal.com/
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
modal voltage will collapse because any change in that
modal reactive power will cause infinite modal voltage
variation.
In (8), let ΔQ = ek where ek has all its elements
zero except the kth
one being 1. Then,
∑
(12)
k th element of
V –Q sensitivity at bus k
∑
∑
(13)
A system is voltage stable if the eigenvalues of the
Jacobian are all positive. Thus the results for voltage
stability enhancement using modal analysis for the
reduced jacobian matrix is when eigen values λ i > 0, the
system is under stable condition eigen values λ i < 0 , the
system is unstable eigen values λi = 0 , the system is
critical and collapse state occurs.
3.0 PROBLEM FORMULATION
The optimal reactive power dispatch problem is
formulated as an optimization problem in which a specific
objective function is minimized while satisfying a number
of equality and inequality constraints. The objectives of
the reactive power dispatch problem considered here is to
minimize the system real power loss and maximize the
static voltage stability margins (SVSM). This objective is
achieved by proper adjustment of reactive power variables
like generator voltage magnitude ( gi ) V , reactive power
generation of capacitor bank (Qci), and transformer tap
setting (tk).Power flow equations are the equality
constraints of the problems, while the inequality
constraints include the limits on real and reactive power
generation, bus voltage magnitudes, transformer tap
positions and line flows. This objective function is
subjected to the following constraints:
3.1Minimization of Real Power Loss
It is aimed in this objective that minimizing of the real power loss (Ploss) in transmission lines of a power
system. This is mathematically stated as follows.
∑
(14)
Where n is the number of transmission lines, gk is the
conductance of branch k, Vi and Vj are voltage magnitude
at bus i and bus j, and θij is the voltage angle difference
between bus i and bus j.
3.2 Minimization of Voltage Deviation
It is aimed in this objective that minimizing of the
Deviations in voltage magnitudes (VD) at load buses. This
is mathematically stated as follows.
Minimize VD = ∑ | | (15)
Where nl is the number of load busses and Vk is the
voltage magnitude at bus k.
3.3System Constraints
In the minimization process of objective functions, some
problem constraints which one is equality and others are inequality had to be met. Objective functions are subjected
to these constraints shown below.
Load flow equality constraints:
– ∑
[
]
(16)
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
∑
[
]
(17)
where, nb is the number of buses, PG and QG are the real
and reactive power of the generator, PD and QD are the
real and reactive load of the generator, and Gij and Bij are
the mutual conductance and susceptance between bus i
and bus j.Generator bus voltage (VGi) inequality
constraint:
(18)
Load bus voltage (VLi) inequality constraint:
(19)
Switchable reactive power compensations (QCi)
inequality constraint:
(20)
Reactive power generation (QGi) inequality constraint:
(21)
Transformers tap setting (Ti) inequality constraint:
(22)
Transmission line flow (SLi) inequality constraint:
(23)
Where, nc, ng and nt are numbers of the switchable
reactive power sources, generators and transformers.
During the simulation process, all constraints satisfied as
explained below [15].
The load flow equality constraints are satisfied by Power
flow algorithm. The generator bus voltage (VGi), the
transformer tap setting (Ti) and the Switchable reactive
power Compensations (QCi) are optimization variables
and they are self-restricted between the minimum and
maximum value by the GSA algorithm *The limits on
active power generation at the slack bus(PGs), load bus
voltages (VLi) and reactive power generation (QGi),
transmission line flow (SLi) are state variables. They are
restricted by adding a penalty function to the objective
functions.
Where
NB number of buses in the system
Ng number of generating units in the system
tk tap setting of transformer branch k
Psl real power generation at slack bus
Vi voltage magnitude at bus i
Pi,Qi real and reactive powers injected at bus i
Pgi,Qgi real and reactive power generations at bus i
Gij,Bij mutual conductance and susceptance between bus i and j
Gii,Bii self-conductance and susceptance of bus i
θij voltage angle difference between bus i and j
4.0 Shuffled frog-leaping algorithm
The shuffled frog-leaping algorithm is a memetic
metaheuristic that is designed to seek a global optimal
solution by performing a heuristic search. It is based on
the evolution of memes carried by individuals and a global
exchange of information among the population (Eusuff
and Lansey 2003). In essence, it combines the benefits of
the local search tool of the particle swarm optimization
(Kennedy and Eberhart 1995), and the idea of mixing
information from parallel local searches to move toward a
global solution (Duan et al. 1993). The SFL algorithm has
been tested on several combinatorial problems and found
to be efficient in finding global solutions (Eusuff and
Lansey 2003). The SFL algorithm involves a population of possible solutions defined by a set of frogs (i.e. solutions)
that is partitioned into subsets referred to as memeplexes.
The different memeplexes are considered as different
cultures of frogs, each performing a local search. Within
each memeplex, the individual frogs hold ideas, that can
be influenced by the ideas of other frogs, and evolve
through a process of memetic evolution. After a number of
memetic evolution steps, ideas are passed among
memeplexes in a shuffling process (Liong and
Atiquzzaman 2004). The local search and the shuffling
processes continue until convergence criteria are satisfied
(Eusuff and Lansey 2003).
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
Pseudo code for SFL algorithm
Begin;
Generate random population of P solutions
(individuals);
For each individual i E P: calculate fitness (i);
Sort the whole population P in descending order of their fitness;
Divide the population P into m memeplexes;
For each memeplex;
Determine the best and worst individuals;
Improve the worst individual position using Eqs. 24and
25;
Repeat for a specific number of iterations;
End;
Combine the evolved memeplexes;
Sort the population P in descending order of their
fitness;
Check if termination=true;
End;
First, an initial population of „P‟ frogs is created
randomly. For S-dimensional problems, each frog i is
represented by S variables as Xi=(xi1, xi 2, . . . . . . , xiS).
The frogs are sorted in a descending order according to
their fitness. Then, the entire population is divided into m
memeplexes, each containing n frogs (i.e. P=m6n). In this
process, the first frog goes to the first memeplex, the
second frog goes to the second memeplex, frog m goes to
the mth memeplex, and frog m+1 goes to the first
memeplex, and so on. Within each memeplex (figure 1b),
the frogs with the best and the worst fitness are identified
as Xb and Xw, respectively. Also, the frog with the global
best fitness is identified as Xg. Then, an evolution process
is applied to improve only the frog with the worst fitness (i.e. not all frogs) in each cycle. Accordingly, the position
of the frog with the worst fitness is adjusted as follows:
Change in frog position (Di ) = rand () .(Xb -Xw)
(24)
New position Xw = current position Xw + Di ; (Dmax ≥ Di
≥ - Dmax )
(25)
where rand( ) is a random number between 0 and 1; and
Dmax is the maximum allowed change in a frog‟s
position. If this process produces a better frog (solution), it
replaces the worst frog. Otherwise, the calculations in
equations (24) and (25) are repeated with respect to the
global best frog (i.e. Xg replaces Xb). If no improvement becomes possible in this latter case, then a new solution is
randomly generated to replace the worst frog with another
frog having any arbitrary fitness (as shown in figure 1b).
The calculations then continue for a specific number of
evolutionary iterations within each memeplex (Eusuff and
Lansey 2003). The main parameters of the SFL algorithm
are: number of frogs P, number of memeplexes, and
number of evolutionary iterations for each memeplex
before shuffling.
4.1 A modified shuffled frog-leaping algorithm
In the SFL algorithm, each memeplex is allowed to evolve
independently to locally search at different regions of the
solution space. In addition, shuffling all the memeplexes
and re-dividing them again into a new set of memeplexes results in a global search through changing the information
between memeplexes. As such, the SFL algorithm
attempts to balance between a wide search of the solution
space and a deep search of promising locations that are
close to a local optimum.
As expressed by equation (24), each individual frog
(solution) in a memeplex is trying to change its position
towards the best frog within the memeplex or the overall
best frog. As shown in this equation, when the difference
in position between the worst frog Xw (i.e. the frog under
evolution) and the best frogs (Xb or Xg) becomes small,
the change in frog Xw‟s position will be very small, and
thus it might stagnate at a local optimum and lead to
premature convergence. To overcome such an occurrence,
this study proposes that the right-hand side of equation
(24) be multiplied by a factor C called the „search –
acceleration factor‟, as follows:
Change in frog position (Di) = rand() . C. ( Xb – Xw)
(26)
Assigning a large value to the factor C at the beginning of
the evolution process will accelerate the global search by
allowing for a bigger change in the frog‟s position and
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Lenin et. al., A MODIFIED SHUFFLED FROG-LEAPING OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM
accordingly will widen the global search area. Then, as the
evolution process continues and a promising location is
identified, the search – acceleration factor, C, will focus
the process on a deeper local search as it will allow the
frogs to
change its positions. The search – acceleration factor, which can be a positive constant value, linear, or nonlinear
function of time, provides the means to balance between
global and local search.
1. Start
2. Determine Population size (p),Number of
memeplexes (m) Iterations within each
memeplex
3. Generate population (p) randomly
4. Evaluate the fitness of (p)
5. Sort (p) in descending order
6. Partition p into m memeplexes
7. Shuffle the memeplexes
8. Is Convergence criteria satisfied?
If yes determine the best solution
If no go back to step 5
9. End
Fig 1. (a) Algorithm for the shuffled frog-leaping
algorithm.
In order to intensify the search, the algorithm has
been modified as follows
1. When m=m+1, it=it+1 then determine
xb, xw , xg.
2. Apply equations 24 , 25
3. Is new frog is better than worst?
a. If no apply equations 24, 25 with replacing
xb by xg.
b. If yes go to step 5 .
4. Is new frog better than worst?
a. If no generate new frog randomly.
b. If yes go to step 5.
5. Replace worst frog
6. End
7. Else go back to determine m and it again
where m = no of memeplexes
It = no of iterations
Fig 1. (b) Algorithm for the shuffled frog-leaping
algorithm
5.0 SIMULATION RESULTS
The validity of the proposed Algorithm technique is
demonstrated on IEEE-30 bus system. The IEEE-30 bus
system has 6 generator buses, 24 load buses and 41
transmission lines of which four branches are (6-9), (6-10)
, (4-12) and (28-27) - are with the tap setting transformers. The real power settings are taken from [1]. The lower
voltage magnitude limits at all buses are 0.95 p.u. and the
upper limits are 1.1 for all the PV buses and 1.05 p.u. for
all the PQ buses and the reference bus.
TABLE 1. Voltage Stability under Contingency State
Sl.No Contigency ORPD
Setting
Vscrpd
Setting
1 28-27 0.1400 0.1422
2 4-12 0.1658 0.1662
3 1-3 0.1784 0.1754
4 2-4 0.2012 0.2032
TABLE 2. Limit Violation Checking of State Variables
State
variables
limits ORPD VSCRPD
Lower upper
Q1 -20 152 1.3422 -1.3269
Q2 -20 61 8.9900 9.8232
Q5 -15 49.92 25.920 26.001
Q8 -10 63.52 38.8200 40.802
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Q11 -15 42 2.9300 5.002
Q13 -15 48 8.1025 6.033
V3 0.95 1.05 1.0372 1.0392
V4 0.95 1.05 1.0307 1.0328
V6 0.95 1.05 1.0282 1.0298
V7 0.95 1.05 1.0101 1.0152
V9 0.95 1.05 1.0462 1.0412
V10 0.95 1.05 1.0482 1.0498
V12 0.95 1.05 1.0400 1.0466
V14 0.95 1.05 1.0474 1.0443
V15 0.95 1.05 1.0457 1.0413
V16 0.95 1.05 1.0426 1.0405
V17 0.95 1.05 1.0382 1.0396
V18 0.95 1.05 1.0392 1.0400
V19 0.95 1.05 1.0381 1.0394
V20 0.95 1.05 1.0112 1.0194
V21 0.95 1.05 1.0435 1.0243
V22 0.95 1.05 1.0448 1.0396
V23 0.95 1.05 1.0472 1.0372
V24 0.95 1.05 1.0484 1.0372
V25 0.95 1.05 1.0142 1.0192
V26 0.95 1.05 1.0494 1.0422
V27 0.95 1.05 1.0472 1.0452
V28 0.95 1.05 1.0243 1.0283
V29 0.95 1.05 1.0439 1.0419
V30 0.95 1.05 1.0418 1.0397
TABLE 3. COMPARISON OF REAL POWER
LOSS
Method
Minimum loss
Evolutionary programming[16]
5.0159
Genetic algorithm[17]
4.665
Real coded GA with Lindex as
SVSM[18]
4.568
Real coded genetic algorithm[19]
4.5015
Proposed MSFL 4.4991
6.0 CONCLUSION
In this paper a novel approach MSFL algorithm used to
solve optimal reactive power dispatch problem,
considering various generator constraints, has been
successfully applied. The proposed method formulates
reactive power dispatch problem as a mixed integer non-
linear optimization problem and determines control
strategy with continuous and discrete control variables
such as generator bus voltage, reactive power generation
of capacitor banks and on load tap changing transformer
tap position. To handle the mixed variables a flexible
representation scheme was proposed. The performance of
the proposed algorithm demonstrated through its voltage stability assessment by modal analysis is effective at
various instants following system contingencies. Also this
method has a good performance for voltage stability
Enhancement of large, complex power system networks.
The effectiveness of the proposed method is demonstrated
on IEEE 30-bus system
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K. Lenin has received his B.E., Degree,
electrical and electronics engineering in
1999 from university of madras, Chennai,
India and M.E., Degree in power systems
in 2000 from Annamalai University, TamilNadu, India. At
present pursuing Ph.D., degree at JNTU, Hyderabad,India.
Bhumanapally. RavindhranathReddy, Born
on 3rd September,1969. Got his B.Tech in
Electrical & Electronics Engineering from
the J.N.T.U. College of Engg., Anantapur in
the year 1991. Completed his M.Tech in
Energy Systems in IPGSR of J.N.T.University Hyderabad
in the year 1997. Obtained his doctoral degree from
JNTUA,Anantapur University in the field of Electrical
Power Systems. Published 12 Research Papers and
presently guiding 6 Ph.D. Scholars. He was specialized in
Power Systems, High Voltage Engineering and Control
Systems. His research interests include Simulation studies
on Transients of different power system equipment.
M. Surya Kalavathi has received her
B.Tech. Electrical and Electronics
Engineering from SVU, Andhra Pradesh,
India and M.Tech, power system operation
and control from SVU, Andhra Pradesh,
India. she received her Phd. Degree from JNTU,
hyderabad and Post doc. From CMU – USA. Currently
she is Professor and Head of the electrical and electronics
engineering department in JNTU, Hyderabad, India and
she has Published 16 Research Papers and presently
guiding 5 Ph.D. Scholars. She has specialised in Power
Systems, High Voltage Engineering and Control Systems.
Her research interests include Simulation studies on
Transients of different power system equipment. She has
18 years of experience. She has invited for various lectures
in institutes.
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