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: gklenin@gmail.com

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-