A Modified Shuffled Frog-Leaping Optimization Algorithm for Solving Optimal Reactive Power Dispatch...

International Electrical Engineering Journal (IEEJ) Vol. 4 (2013) No. 2, pp. 1049-1058 ISSN 2078-2365 http://www.ieejournal.com/

1049

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

mailto:[email protected]


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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.


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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


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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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∑

[

]

(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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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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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., Test driving three 1995 genetic algorithms: new test

functions and geometric matching. J. Heuristics, 1995, 1,

77 – 104.

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.

A Modified Shuffled Frog-Leaping Optimization Algorithm for Solving Optimal Reactive Power Dispatch...

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