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Hybrid Eagle Strategy Flower Pollination
Algorithm for Solving Optimal Reactive Power
Dispatch Problem
K. Lenin and B. R. Reddy Jawaharlal Nehru Technological University Kukatpally, Hyderabad 500 085, India
Email: [email protected]
Abstract—The prime aspect of solving Optimal Reactive
Power Dispatch Problem (ORPD) is to minimize the real
power loss and also to keep the voltage profile within the
limits. For any metaheuristic search algorithm, it is very
significant to poise exploration and exploitation because the
communication of these two key mechanisms can drastically
affect the efficiency of the search. To explore this
challenging issue by using eagle strategy in combination
with flower algorithm has been designed. The proposed
Hybrid Eagle Strategy Flower Pollination algorithm
(HESFPA) has been validated, by applying it on standard
IEEE 30 bus test system. The results have been compared to
other heuristics methods and the proposed algorithm
converges to best solution.
Index Terms—eagle strategy, flower algorithm, optimization,
nature-inspired algorithm, metaheuristic, optimal reactive
power, transmission loss
I. INTRODUCTION
Reactive power optimization plays an important task in
optimal operation of power systems. Many papers by
various authors has been projected to solve the ORPD
problems such as, gradient based optimization algorithm
[1], [2], quadratic programming, non linear programming
[3] and interior point method [4]-[7]. In recent years,
standard genetic algorithm (SGA) [8], the adaptive
genetic algorithm (AGA) [9], and partial swarm
optimization PSO [10], [11] have been applied for
solving ORPD problems. Due to the problem of
unmatched generation and transmission capability growth
and due to continuous increase in demand of electrical
power the ORPD problem has become very complicated.
The incapability of the power system to meet the demand
for reactive power to preserve regular voltage profile in
stressed situations is playing very significant role for
causing voltage collapse. In the past many innovative
algorithms such an Evolutionary Algorithm [12], [13],
Genetic algorithm [14], [15], Evolutionary strategies
[16]-[18], Differential Evolution [19], [20], Genetic
programming [21] and Evolutionary programming [22]
are used to solve many rigid problems in optimization. In
this research paper both the Eagle strategy and Flower
pollination algorithm has been hybridized to solve the
Manuscript received March 18, 2014; revised June 27, 2014.
ORPD Problems. This algorithm (HESFPA) is applied to
obtain the optimal control variables so as to improve the
voltage stability level of the system. The performance of
the proposed method has been tested on IEEE 30 bus
system and the results are compared with the standard
GA and PSO method.
II. PROBLEM FORMULATION
The Optimal Power Flow problem has been considered
as general minimization problem with constraints, and
can be mathematically written as:
Minimize f(x,u) (1)
Subject to g(x,u)=0 (2)
and
(3)
where f(x,u) is the objective function. g(x.u) and h(x,u)
are respectively the set of equality and inequality
constraints. x is the vector of state variables, and u is the
vector of control variables.
The state variables are the load buses (PQ buses)
voltages, angles, the generator reactive powers and the
slack active generator power:
(4)
The control variables are the generator bus voltages,
the shunt capacitors and the transformers tap-settings:
(5)
or
(6)
where Ng, Nt and Nc are the number of generators,
number of tap transformers and the number of shunt
compensators respectively.
III. OBJECTIVE FUNCTION
A. Active Power Loss
The goal of the reactive power dispatch is to minimize
the active power loss in the transmission network, which
can be mathematically described as follows:
International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
©2014 Engineering and Technology Publishing 221doi: 10.12720/ijoee.2.3.221-225
(7)
or
(8)
where gk is the conductance of branch between nodes i
and j, Nbr is the total number of transmission lines in
power systems. Pd is the total active power demand, Pgi is
the generator active power of unit i, and Pgsalck is the
generator active power of slack bus.
B. Voltage Profile Improvement
For minimization of the voltage deviation in PQ buses,
the objective function formulated as:
(9)
Where ωv: is a weighting factor of voltage deviation.
VD is the voltage deviation given by:
(10)
C. Equality Constraint
The equality constraint g(x,u) of the ORPD problem is
represented by the power balance equation, where the
total power generation must envelop the total power
demand and the power losses:
(11)
D. Inequality Constraints
The inequality constraints h(x,u) imitate the limits on
components in the power system as well as the limits
created to guarantee system security. Upper and lower
bounds on the active power of slack bus, and reactive
power of generators:
(12)
(13)
Upper and lower bounds on the bus voltage
magnitudes:
(14)
Upper and lower bounds on the transformers tap ratios:
(15)
Upper and lower bounds on the compensators reactive
powers:
(16)
where N is the total number of buses, NT is the total
number of Transformers; Nc is the total number of shunt
reactive compensators.
IV. INTERMITTENT SEARCH THEORY
Normally exploitation tends to increase the speed of
convergence, while exploration tends to decrease the
convergence rate of the algorithm. Also too much
exploration increases the chance of finding the global
optimality but with a reduced efficiency, but well-built
exploitation tends to make the algorithm being trapped in
a local optimum. Therefore, there should be a fine
balance between the precise amount of exploration and
the right level of exploitation. Some algorithms may have
essentially better balance among these two prime
components than other algorithms, and that is also one of
the reasons why a quantity of algorithms may perform
better than others. Intermittent search strategy
apprehension an iterative strategy consisting of a slow
phase and a fast phase [23], [24]. Let a and R be the
mean times used up in intensive detection stage and the
time used up in the exploration stage, respectively, in the
2D case [23]. The diffusive search process is govern by
the mean first-passage time satisfying the following
equations
(17)
(18)
where t1 and t2 are times spent during the search process
at slow and fast stages, respectively, and u is the average
search speed [24]. After some extensive numerical
examination [23], [24], the optimal balance of these two
stages can be estimated as
(19)
V. EAGLE STRATEGY
Eagle strategy is a metaheuristic approach for
optimization, developed in 2010 by Xin-She Yang and
Suash Deb [25]. It uses a mixture of crude global search
and intensive local search. In essence, the strategy first
explores the search space globally using a Levy flight
random walk, if it finds a promising solution, then a
concentrated local search is employed using a well-
organized local optimizer such as hill-climbing,
differential evolution and algorithms. Then, the two-stage
process starts again with new comprehensive exploration
followed by a local search in a new area. In this strategy
mainly pe controls the switch between local and global
search.
Algorithm of the eagle strategy
Step 1. Objective functions f(x)
Step 2. Initialization and random initial
presumption xt=0
Step 3. while (stop criterion)
Step 4. Global exploration by randomization
Step 5. Evaluate the objective
Step 6. If pe<rand, switch to a local search
Step 7. Intensive local search around a capable
solution through an well-organized optimizer
Step 8. if (a better solution is found)
Step 9. Update the current best
end
end
Step 10. Update t=t+1
International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
©2014 Engineering and Technology Publishing 222
end
Step 11. Post-Route the results and revelation.
VI. FLOWER POLLINATION ALGORITHM
Flower pollination algorithm (FPA), or flower
algorithm, was developed by Xin-She Yang in 2012 [26],
inspired by the flow pollination process of flowering
plants.
We use the following systems in FPA,
System 1. Biotic and cross-pollination has been
treated as global pollination process.
System 2. For local pollination, A- biotic and self-
pollination has been used.
System 3. Pollinators such as insects can develop
flower reliability, which is equivalent to a
reproduction probability and it is proportional to
the similarity of two flowers implicated.
System 4. The communication of local pollination
and global pollination can be controlled by a
control probability p∈[0, 1], with a slight bias
towards local pollination.
System 1 and flower reliability can be represented
mathematically as
(20)
where is the pollen i or solution vector xi at iteration t,
and g* is the current best solution found among all
solutions at the current generation/iteration. Here γ is a
scaling factor to control the step size. L(λ) is the
parameter that corresponds to the strength of the
pollination, which essentially is also the step size. Since
insects may move over a long distance with various
distance steps, we can use a Levy flight to mimic this
characteristic efficiently. We draw L>0 from a Levy
distribution
(21)
here, Γ(λ) is the standard gamma function, and this
distribution is valid for large steps s>0.
Then, to model the local pollination, for both system 2
and system 3 can be represented as
(22)
where and are pollen from different flowers of the
same plant species. This essentially mimics the flower
reliability in a limited neighbourhood. Mathematically, if
and comes from the same species or selected from
the same population, this equivalently becomes a local
random walk if we draw from a uniform distribution in
[0,1].
Flower Pollination Algorithm
Step 1. Objective min of (x), x = (x1, x2, ..., xd)
Step 2. Initialize a population of n flowers
Step 3. Find the best solution g* in the initial
population
Step 4. Define a control probability p ∈ [0, 1]
Step 5. Define a stopping criterion (a fixed number
of generations/iterations)
Step 6. while (t<Max Generation)
Step 7. for i = 1:n (all n flowers in the population)
Step 8. if rand<p,
Step 9. Draw a (d-dimensional) step vector L
which obeys a Levy distribution Global
pollination through
else
Step 10. Draw from a uniform distribution in
[0,1]
Step 11. Do local pollination through
end if
Step 12. Evaluate new solutions
Step 13. If new solutions are better, update them in
the population
end for
Step 15. Find the current best solution g*
end while
Output - best solution has been found
VII. PROCEDURE OF HYBRID EAGLE STRATEGY
FLOWER POLLINATION ALGORITHM FOR
SOLVING OPTIMAL REACTIVE POWER DISPATCH
PROBLEM
As Eagle Strategy is a two-stage strategy, we can
employ different algorithms at different phase. The large
extent search stage can use randomization via Levy
flights. The Levy distribution is an allocation of the sum
of n identically and independently allocation random
variables. For the second phase, we use differential
evolution as the intensive local search. FPA is a global
search algorithm; it can easily be tuned to do efficient
local search by limiting new solutions locally around the
most capable region and is achieved by setting γ to a very
little value. The sense of balance of local search and
global search is very significant, and so is the balance of
the first stage and second stage in the Eagle Strategy. For
more exploration, in flower algorithm, normally we use
p=0.9, γ=0.1 and λ=1.5 for the applications here. In terms
of balancing the exploration and exploitation in the
combination of eagle strategy and flower algorithm, the
most important parameter is pe in the eagle strategy. For
isotropic random walks for local exploration, we have
D≈s2/2, where s is the step length with a jump during a
unit time interval or each iteration step. From equation
(19), the optimal ratio of exploitation and exploration in a
special case of R/a1, and we can write
(23)
Exploration stage should have more time for the
intensive search of solution.
VIII. SIMULATION RESULTS
HESFPA algorithm has been tested on the IEEE 30-
bus, 41 branch system. It has a total of 13 control
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©2014 Engineering and Technology Publishing 223
variables as follows: 6 generator-bus voltage magnitudes,
4 transformer-tap settings, and 2 bus shunt reactive
compensators. Bus 1 is the slack bus, 2, 5, 8, 11 and 13
are taken as PV generator buses and the rest are PQ load
buses. The variables limits are listed in Table I.
TABLE I. INITIAL VARIABLES LIMITS (PU)
Control
variables
Min. value Max. value Type
Generator: Vg 0.90 1.10 Continuous
Load Bus: VL 0.95 1.05 Continuous
T 0.95 1.05 Discrete
Qc -0.12 0.36 Discrete
The transformer taps and the reactive power source
installation are discrete with the changes step of 0.01. The
power limits generators buses are represented in Table II.
Generators buses are: PV buses 2,5,8,11,13 and slack bus
is 1.the others are PQ-buses.
TABLE II. GENERATORS POWER LIMITS IN MW AND MVAR
Bus n° Pg Pgmin Pgmax Qgmin
1 98.00 51 202 -21
2 81.00 22 81 -21
5 53.00 16 53 -16
8 21.00 11 34 -16
11 21.00 11 29 -11
13 21.00 13 41 -16
TABLE III. VALUES OF CONTROL VARIABLES AFTER OPTIMIZATION
AND ACTIVE POWER LOSS
Control
Variables (p.u)
HESFPA
V1 1.0541
V2 1.0490
V5 1.0294
V8 1.0390
V11 1.0809
V13 1.0572
T4,12 0.00
T6,9 0.03
T6,10 0.90
T28,27 0.90
Q10 0.11
Q24 0.11
PLOSS 4.2936
VD 0.8980
The proposed approach succeeds in maintenance the
dependent variables within their limits as shown in Table
III. Table IV summarize the results of the optimal
solution obtained by PSO, SGA and HESFPA methods. It
reveals the decrease of real power loss after optimization.
TABLE IV. COMPARISON RESULTS
HESFPA
4.98 Mw 4.9262Mw 4.2936Mw
IX. CONCLUSION
In this paper, the proposed HESFPA has been
successfully implemented to solve ORPD problem. The
main advantage of the algorithm is solving the objective
function with real coded of both continuous, discrete
control variables, and easily handling nonlinear
constraints. The proposed algorithm has been tested on
the IEEE 30-bus system. And the results were compared
with the other heuristic methods such as SGA and PSO
algorithm reported in the literature.
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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. He presently is pursuing Ph.D. degree at JNTU, Hyderabad, India.
Bhumanapally Ravindhranath Reddy, born
on 3rd September,1969. Got his B.Tech in
Electrical and Electronics Engineering from the J.N.T.U. College of Engg., Anantapur in
the year 1991. He completed his M.Tech in Energy Systems in IPGSR of J.N.T.University
Hyderabad in the year 1997. He obtained his
doctoral degree from JNTUA, Anantapur University in the field of Electrical Power
Systems. He published 12 research papers and presently is 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.
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2012, pp. 240-