load flow solution using hybrid particle swarm optimization.pdf

8/10/2019 Load flow solution using hybrid particle swarm optimization.pdf

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Now the optimization problem can be formulatedas

follows:

Minimize f ( v,6 ) (11)

Subject to

Vslack scheduled value of the slack bus voltage

V,

=

scheduled value of the PV-bus voltage

P, = scheduled value of PV-bus generated power

P NPV

Npv: set of PV-buses

Where f =

C Fpi

+ Fq,

) (12)

So

the objective function of the load flow is to

minimize these functio ns to get the voltages and angles

of the buses, which satisfy the power balance equations.

111.Hybrid Particle Swarm Optimization HPSO)

Th e particle swarm optimization algorithm was

originally introduced by Kennedy and Eberhart in 1995

as an alternative to the standard Genetic algorithm

(GA). The PSO was inspired by insect swarms and has

since proven to be a competitor to the GA whenit

comes to function optimization. The PSO model

consists of a numb er of particles moving around in the

search space, each representing a possible solution to a

numerical problem. Each particle has a position vector

(xi) and a velocity vector y ), the position Mi)s the

best position encountered by the particle (i) during its

search and the position gbest) s that of the best

particle in the s warm group.

In each iteration the velocity of each particle is updated

according to its best-encountered position and the best

position encountered amon g the group , using the

following equation:

where

:

known as the constriction coefficient

w

:

nertia weight

3 , a 2

:

are random values different for each

particle and for each dimension between [0,2]

Th e position o f each particle is then updated in each

iteration by adding the velocity vector to the position

vector.

Equation (13) consists of three terms: the first one

the second

best previous and current position. Finally, the effect of

the swarm group best experience on the velocity of

each individual in the group. This effect is considered

in equation (13

experie nce (the position of the best particle in the

swarm group) and the i-th

on.

Equation (14) simulates the flying of the particle

toward a new position. The role of the inertia weight w

is considered very important in PSO convergence

behavior [6]. The inertia weight is employed to control

the impact of the previous history of velocities on the

current velocity. In this way, the parameter w regulate s

the trade-off betw een the global and local explor ation

abilities of the swarm. A large inertia weight facilitates

global exploration (searching new areas), while a small

one tends to facilitate local exploration, i.e. fine-tun ing

the curren t search area. A suitable value for the inertia

weight w usually provides balance between global and

local exploration abilities and consequently a reductio n

on the number of iterations required to locate the

optimum solution. T here has been a lot of research in

how to improve the performance of the PSO in means

of faster convergence and to make sure that the PS O

will n ot get stuck in a local minim a [7]-[9]. Th e

improvements in the P SO are done by trying to have

some of the properties as in the GA besi de the PSO

own properties. One of the most powerful proper ties of

the GA is the ability to breed and produce better

individuals (children) than the old ones (parents). Th is

technique is used in the algorithm proposed by this

work. It is used to accelerate the solution of the

problem.

A

hybrid model of the standard

GA

a nd the PSOis

introduced in [lo] . This model incorporates one major

aspect of the standard GA into the PSO , which is the

reproduction or breeding. Breeding is one of the core

elements that make the standard G A a powerful

algorithm. Therefore, a hybrid PS O with the breeding

property has the potential to reach a better optim um

than the standard PSO. The model for the breeding

process is as mentioned in [lo ]:

For the positions of the children:

childl(xi)

=

pi

*

parentl(xi)+ 1- pi)

*

parentz(xi)

child2(xi)

=

pi parent2(xi)

+

(1- pi)

*

parentl(xi)

For the velocity vectors of the ch ildren:

childl@)

=

(parentl@)

+

parent2@))

*

Iparentl(y)l /

child2(v) = (parentl(v) + parent2(v)) * Iparentz(v)l

(15)

(16)

Iparentl@) + parentz(y)l (17)

Iparentl&) + parentz(y)l (18)

where pi is a uniformly distributed random number

between [0,1]

parentl(xi) : position vector of a randomly chosen

particle to take part in the breeding process.

parent2(xi) : position vector of a randomly chosen

particle to be the other parent in the breeding process.

childl(xi) : position vector

of

the first offspring

childz(xi)

:

position vector of the first offspring

parentl&)

:

velocity vector of the first parent

parent2@): velocity vector of the second parent

IV.Numerical Examples

In this section, the Ward-Hale 6-bus and the IEE E 14-

bus systems are u sed to show the applicability of the

proposed algorithm.

It is well known, as shown in the P-V curve fig. ( l ) ,

that at any given load on the system there are two

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solutions for the load flow

[

111. One is a stable solution

(at point 1) and the other one is unstable (at point 2)

from the v oltage stability point of view. This mean s

that as an optimization problem the LF has two global

minima. T o avoid getting to the unstable global

minima, the initial positions of the particles were

chosen to be in the neighborhood of the stable solution.

This was accomplished by initializing the voltages

randomly in the range of 0.95

This approach gave good results and the PSO reached

the global minimum that is stable.

1.05 pu.

3 -0.55 -0.13

4

-0.0

-0.0

5

-0.3 -0.18

6 -0.5 -0.05

I

I

0.0

0.0

0.0

0 0

System

loading

Fig.

1)

P-V

c urve

a) Ward-Hale

6-bus

system:

The W ard-Hale 6-bus system consists of two

generators, four load buses, and seven branches of

which two branches (2 -3,4 -5) are under load tap setting

transformers. Th e system is show n in figure (2). The

loads are given in table (1). Th e branch parameters are

given in table (2). Generator no. 1 is set as the slack bus

with voltage magnitude

=

1.05 pu. The second

generator is a PV -bus w ith sched uled voltage

magnitude = 1. pu and scheduled generated power =

0.5

pu. Th e setting of the LTC (2-3) is 0.909, while the

setting of the one (4-5) is 0.975.The parameters used in

the PSO model were as follows: number of particles =

100, w =1.075. Th e constriction factor was decreasing

with the number of iterations as follows:

=

0.95

=

0.94 k>400

0.935 k%OO

=

0.93 k> 14

where k is the iteration counter.

The stopping criterion used was some specified

tolerance

for

the maximum Fpi r Fqi

This tolerance w as 0.0001. T he algorithm solved the

problem after (8 51) iterations with a tolerance of

(8.1644*10-5).

To

prove the effectiveness and

robustness of the method the loading of the system was

changed. The algorithm w as also able to reach the

specified tolerance. Other parameters in the network

such the voltage magnitude of the slack bus, the voltage

or power generated at a PV-bus or the settings of the

were also changed and the proposed method

successfully solved the problem.

0.0

Node 1: is the slack bus.

SC: stands for shun t capacitors installed at this bus.

Table (2) Branch data

Bus Impedance Halfofl ine

9

4

A 3

I 3

2

Fig.

2) Ward-Hale 6-bussystem

b) IEEE 14-bussystem:

The network shown in figure (3), consists of two

generators, 12 load buses. Three of these load buses are

P-V buses beside one of the generators. The other

generator is taken as the slack bus. The data ofthe

network is found in [9]. The operating conditions of the

system are shown in table (2). Th e algorithm was also

able to solve this system using the same parameters as

before. The algorithm needed about (4994) iterations to

reach a tolerance of (9.3366*10-5).

V.

Analysis

of

the Resul ts

The proposed algorithm was capable of solving the load

flow problem with the required tolerance. It was

successful in m any cas es where the load distribution

among the buses was changed both active and reactive

powers. It also solved the problem when the settings of

the slack bus, the PV -buses were changed. As a

comparison between the solution obtained by PSO

against the solution by Newton -Raphson technique,

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table (4 ) shows both solutions. Another ca se was solved

where the system was near to its m aximum loading

point of the system. This maximum loading was

obtained by the continuation power flow [2]. Therefore,

the PSO was able

to

solve the problem even near the

maxim um loading point where the NR technique may

sometimes fail

to

solve the problem due to the

singularity of the Jaco bian matrix.

9

10

11

12

13

14

4 4

-0.295 -0.166 0.19

-0.090 -0.058 0.0

-0.035

-0.018

0.0

-0.061 -0.016 0.0

-0.135

-0.058

0.0

-0.149 -0.05

0.0

Fig. 3)

IEEE

14-bussystem

Ta m

Nod e 1: is the slack bus.

SC: stands fo r shunt capacitors installed at this bus.

VLConclusions

A new application for particle swarm optim ization has

been developed. Th e PSO algorithm has been

strengthened using breeding technique similar to that

applied in Genetic algorithm (GA). Th e new suggested

algorithm has been applied to two test system s (Hale-

Ward system and the IEE E 14-bus system). Th e results

proved the applicability and validity of the new

algorithm as a new

tool

for load flow solution that

could be helpful in other studies when problems are

encountered due to Jacobian singularity in the classical

techniques.

References

1

Glem

W.

Stagg, Ahmed H. Alabiad

meth ods in Pow er system analysis Macgraw-Hill,

1968.

2. V. Ajjarapu

flow: A tool for steady state voltage stability

7,

No. 1, February 1992

Optimization

Networks, page pp. 19421948,1995

3. J. K ennedy and R . Eberhart

In Proc. IEEE Int. Conf. Neural

4. Kit Po Wong, An Li he Load flow

Computation 1995, IEEE International C onference

on, Volume 1, 29 Nov.- 1 Dec. 1 995

5.

Y. Fukuyama,

S.

Takayama, H.Yoshida,

KKawata, and Y. Nakanishi

optimization for reactive power and voltage control

Trans. on Power Systems, pages 123 2-123 9,200 0

6

Y. Shi and R. Eberhart

Proc. P ~ n n .

Conf. Evolutionary Program.,

Mar.

1998, pp. 591-

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7.

G. Ciuprina, D. Ioan and I. Munteanu

Intelligent-Particle Swarm Optimization in

Electromagnetics Magn., vol. 38,

pp. 1037-1040,2002.

8. Vladimiro M iranda and Nuno Fonseca

Evolutionary Particle Swarm Algorithm (EPSO)

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Sevilla, 24-28 June 2002, Session 21, Paper 5.

Estimation Using Hybrid Particle Swarm

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Hybrid Particle Sw arm Optimizer with Breeding

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Lovbjerg, T. Kiel Rasmussen and

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Condition with Nose Curve Using Homotopy

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