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Conclusion References Keywords: particle swarm optimization, pseudo-gradient search, optimal reactive power dispatch, voltage stability index, optimal power flow, optimization, real powe

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Optimal Reactive Power Dispatch Using Improved Pseudo-gradient Search Particle Swarm Optimization

Jirawadee Polprasert, Weerakorn Ongsakul & Vo Ngoc Dieu

To cite this article: Jirawadee Polprasert, Weerakorn Ongsakul & Vo Ngoc Dieu (2016): Optimal

Reactive Power Dispatch Using Improved Pseudo-gradient Search Particle Swarm Optimization, Electric Power Components and Systems, DOI: 10.1080/15325008.2015.1112449

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ISSN: 1532-5008 print / 1532-5016 online

DOI: 10.1080/15325008.2015.1112449

Optimal Reactive Power Dispatch Using Improved

Pseudo-gradient Search Particle Swarm Optimization

Jirawadee Polprasert,1Weerakorn Ongsakul,1and Vo Ngoc Dieu2

1

Energy Field of Study, School of Environment, Resources and Development, Asian Institute of Technology, Klong Luang, Pathumthani, Thailand

2

Department of Power Systems, Faculty of Electrical and Electronics Engineering, Ho Chi Minh City University of Technology,

Ho Chi Minh City, Vietnam

CONTENTS

1 Introduction

2 ORPD Problem Formulation

3 IPG-PSO

4 Simulation Results

5 Conclusion

References

Keywords: particle swarm optimization, pseudo-gradient search, optimal

reactive power dispatch, voltage stability index, optimal power flow,

optimization, real power loss, voltage deviation

Received 14 August 2013; accepted 9 October 2015

Address correspondence to Dr Weerakorn Ongsakul, Energy Field of Study,

School of Environment, Resources and Development, Asian Institute of

Technology, Klong Luang, Pathumthani, 12120, Thailand E-mail:

ongsakul@ait.asia

Color versions of one or more of the figures in the article can be found online

at www.tandfonline.com/uemp.

Abstract—This article proposes an improved pseudo-gradient

search-particle swarm optimization (IPG-PSO) approach for solving the optimal reactive power dispatch (ORPD) problem This ORPD problem is to determine optimal control variables, such as generator bus voltages, settings of shunt VAR compensators, and tap settings

of on-load tap change (OLTC) transformers, for minimizing the real power loss, voltage deviation, and voltage stability index satisfy-ing power balance equations and generator and network operatsatisfy-ing limit constraints The proposed method is an improved PSO using

a linearly decreasing chaotic inertia weight factor and guided by a pseudo-gradient search, which determines an appropriate direction

of particles toward a global optimal solution The proposed IPG-PSO method is used to minimize three different single-objective functions, including real power loss, voltage deviation, and voltage stability in-dex Test results on the IEEE 30-bus and 118-bus systems indicate that the proposed IPG-PSO method renders a higher solution quality and faster computing time than other methods Accordingly, the pro-posed IPG-PSO for solving ORPD problem is potentially viable for online implementation

1 INTRODUCTION

Optimal reactive power dispatch (ORPD), which is a sub-problem of optimal power flow (OPF), is used to obtain a secure and economic system operation ORPD is to determine optimal settings of control variables, such as generator bus voltage, transformer tap setting, and reactive power of shunt VAR compensator, to minimize the specified objective func-tion satisfying equality and inequality constraints Objective functions include minimization of real power loss, minimiz-ing voltage deviation to improve voltage profile at load buses, and minimizing the voltage stability index to enhance voltage stability [1, 2]

The OPF problem can be separated into two sub-problems:

the P-problem and Q-problem A P-problem, called the OPF

problem, is computing optimal real power generation outputs for minimizing the total generation fuel cost while satisfying

1

power balance equations, generation, and network operating

limits A Q-problem, called the ORPD problem fixing real

power generation from the P-problem, is computing optimal

control variables, which are the generator or PV bus voltages,

tap settings of on-load tap-changing transformers, static VAR

compensators for minimizing the transmission losses, voltage

deviation for voltage profile improvement, and the voltage

sta-bility index for voltage stasta-bility enhancement while satisfying

power balance equations, generator, and network operating

limits The control variables of the Q-problem ORPD will be

fixed in the P-problem OPF again The iterative process will

continue until there is no significant difference of control

vari-ables mismatch between the P- and Q-problems ORPD is well

established, but their solutions are still far from optimal

solu-tions There is a need to find a better ORPD solution for online

operation

The ORPD problem has long been solved by various

con-ventional methods [2–9], such as linear programming (LP),

non-linear programming (NLP), the Newton method, quadratic

programming (QP), the interior point method (IP), dynamic

programming (DP), mixed-integer programming (MIP), and

gradient search Nevertheless, these techniques may suffer

from slow convergence, being trapped in local optimality, and

difficulty in handling non-linear discontinuous functions and

constraints In addition, LP and QP methods require

piece-wise linear objective approximation and piecepiece-wise quadratic

objective approximation, respectively

To date, stochastic or meta-heuristic algorithms have

been developed for solving ORPD problems, such as

quasi-oppositional teaching learning based optimization (QOTLBO)

[10], gravitational search algorithm (GSA) [11], differential

evolution (DE) [12], harmony search algorithm (HSA) [13],

genetic algorithm (GA) [14], evolutionary programming (EP),

simulated annealing (SA), biogeography-based optimization

(BBO) [15], ant colony optimization algorithm (ACOA) [16]

Those techniques can handle convex, smooth,

non-differential objective functions and non-linear constraints

Among the meta-heuristic methods, a particle swarm

op-timization (PSO) is a population-based search opop-timization

algorithm for solving the ORPD problem that has better

search-ing ability with faster performance than other meta-heuristic

methods There are many PSO methods for solving the ORPD

problem and optimization problems, such as PSO with

time-varying inertia weight (PSO-TVIW), PSO with time-time-varying

acceleration coefficients (PSO-TVAC), self-organizing PSO

with TVAC (SPSO-TVAC), stochastic weight trade-off PSO

(SWT-PSO), comprehensive learning PSO (CL-PSO) [17], and

pseudo-gradient PSO (PG-PSO), which is applied to the

eco-nomic dispatch problem [18] Many PSO variations can search

for optimal solutions in a shorter computational time In

addi-tion, hybrid methods have been widely employed for solving optimization problems, such as the hybrid multi-agent-based PSO (HMAPSO) [19], hybrid GA [20], hybrid EP [21], and hy-brid PSO [22] These methods usually obtain a higher solution quality than the single methods In [18], a pseudo-gradient search is used to determine an appropriate direction during the search process Nevertheless, these PSO options can be further enhanced by using appropriate control parameters to better guide the search directions

In this article, an improved pseudo-gradient (IPG) search–PSO (IPG-PSO) method is proposed for solving the ORPD problem The proposed method is improved by a dy-namic weight factor using chaotic sequences and a linearly de-creasing inertia weighting factor, which is used to diversify the search space during the early stage of iterations and intensify the search space during the later stage of iterations Addition-ally, the IPG-PSO is guided by a “pseudo-gradient” search to find a better direction of particles so that they can achieve a nearly global optimal solution The proposed IPG-PSO method

is applied to three different single-objective functions mini-mizing real power system loss, voltage deviation at load buses, and the voltage stability index and satisfying power balance equations, generator voltages and reactive power limits, reac-tive power of shunt VAR capacitor compensation limits, trans-former tap setting limits, voltages at load buses, and transmis-sion line loading limits The proposed IPG-PSO method has been tested on the IEEE 30-bus and 118-bus systems with dif-ferent single-objective functions, and the obtained results are compared to many PSO algorithms and other methods reported

in the literature

The rest of the article is organized as follows In Section 2, objective functions and constraints of the ORPD problem are formulated The feature of the proposed IPG-PSO algorithm

is described in Section 3 The simulation results obtained from the proposed IPG-PSO method and comparisons on the IEEE 30- and 118-bus systems are illustrated in Section 4 Finally, a conclusion is given in Section 5

2 ORPD PROBLEM FORMULATION

Mathematically, the ORPD objective functions and constraints can be formulated as given in what follows

2.1 Minimization of Real Power Loss

The first objective is to minimize the real power loss as Minimize F1= P loss (s , u) =

N L



l=1

g l



|V i|2+V j2

− 2 |V|V cos

δ − δ , (1)

FIGURE 1 Characteristics of different types of inertia weight

factor: (a) constant weight factor, (b) linearly decreasing weight

factor, (c) chaotic weight factor, and (d) proposed linearly

decreasing chaotic weight factor

where NL is the number of transmission lines; g l is the

con-ductance of branch l connecting buses i and j; V i and V j are

voltage magnitudes at buses i and j, respectively; δ iandδ jare

voltage phase angles at buses i and j, respectively.

2.2 Minimization of Voltage Deviation at Load Buses

For voltage profile improvement, the second objective is to

minimize voltage deviation at all load buses, defined as

Minimize F2= V D (s, u) =

N L B



i=1

|V i| −V i ,spec , (2)

where NLB is number of load buses, and V i ,spec is a

pre-specified voltage magnitude at load bus i, which is usually set

to 1.0 p.u

2.3 Minimization of Voltage Stability Index

To enhance voltage stability, the third objective function is to

minimize the voltage stability index [23]:

Minimize F3= Lmax(s , u) = max {L k } , (3) where

L k=

1 +V ok

V k



 = S∗k+

Y kk∗+|V k|2





 =











1−

N G

i=1F ki V i

V k











[F ki]= − [Y L L]−1[Y L G]. (5)

It is noted that S k+ consists of two parts, which can be

expressed as

S k+= S k+

⎛

i ∈L

i =k

Z ki∗S i

Z∗kk V i

⎞

⎟

⎠ · V k = S k + S kcorr , (6)

where

S kcorr =

⎛

⎜

i ∈L

i =k

Z ki∗S i

Z∗kk V i

⎞

⎟

V ok = −

N G



i=1

F ki · V i , (8)

Y kk+= 1

Z kk

where

L k is voltage stability index of the kth bus;

Y L L , Y L G are PQ and PV sub-matrices of the admittance matrix (Y bus);

[F ki] is a hybrid matrix that is generated by a partial inversion

of sub-matrices [Y L L], [Y L G]; and

S k+is the transformed apparent power.

In fact, L-index is an effective quantitative measurement for

the system to find how far the current states of the system can

go to the voltage collapse point The L-index at load bus k in

Eq (4) is mainly influenced by the equivalent load, including

the load at bus k itself (S k) and the contributions of the other

load buses (S kcorr ) When the load at bus k is increasing, L k

will be increasingly closer to one or the collapse point Note

that L-index ranges from 0 (no load of system) to 1 (voltage

collapse)

The vector of state and output variables (s), which

con-tain the voltages at load bus, reactive power generations, and transmission line flows, is represented as

s=|V L1 | |V LNLB | , Q G1 Q GNG , S L1 S LNL

T

. (10)

The vector of control or independent variables (u), which

contain the voltages at load bus, reactive power generations, and transmission line flows, is represented as

u = [|V G1 | |V GNG | , T1 T NT , Q c1 Q cNC]T (11)

2.4 System Constraints

2.4.1 Equality Constraints These power flow constraints are to balance the real and re-active power outputs of generator and load and loss, given

Scenario x f δ(x)

TABLE 1 Characteristic of direction indicator of component x i

by

P i = P Gi − P Di = |V i|

NB



j=1

V jG i jcos

δ i − δ j



+ B i jsin

δ i − δ j



, i = 1, , N B, (12)

Q i = Q Gi − Q Di = |V i|

NB



j=1

V jG i jsin

δ i − δ j



− B i jcos

δ i − δ j



, i = 1, , N B, (13) where

NB is the number of buses;

P i and Q i are real and reactive power injection at bus i,

respec-tively;

P Gi and Q Gi are real and reactive power generation at bus i,

respectively;

P Di and Q Di are real and reactive load demand at bus i,

respec-tively;

|V i| andV jare voltage magnitudes at buses i and j,

respec-tively; and

G ij and B ij are the ijth real and imaginary element of Y bus,

respectively

2.4.2 Inequality Constraints

Generator constraints: The generator voltages and generator

reactive power outputs are limited by their minimum and

maximum values as

Vmin

Gi  ≤ |V Gi| ≤Vmax

Gi , i = 1, , N G, (14)

QminGi ≤ Q Gi ≤ Qmax

Gi , i = 1, , N G.

Transformer tap setting constraints: The tap settings are

con-strained by their lower and upper bounds as

T imin≤ T i ≤ Tmax

i , i = 1, , N T. (16)

Shunt VAR compensator constraints: The reactive powers of

shunt VAR compensator devices are bounded by their lower

and upper values as

Qminci ≤ Q ci ≤ Qmax

ci , i = 1, , NC. (17)

Load bus voltage and line constraints: The load bus voltages

and transmission line flows are constrained by the lower and upper operating limits as

Vmin

Li  ≤ |V Li| ≤Vmax

Li , i = 1, , N L B, (18)

|S Li | ≤ Smax

Li , i = 1, , N L, (19)

where Q ci is the reactive power of the shunt VAR

com-pensator at unit i, V Li is the voltage at load bus i, NG is the number of generators, NT is the number of tap setting trans-former branches, and NC is the number of shunt capacitor

compensators

3 IPG-PSO

PSO is inspired by animal social behavior, such as flocking

of birds and schooling of fish, which are called particles Par-ticles will roam in the search space and move together for finding the optimal solution in which they will be sharing in-formation More specifically, each particle in the group in the

d-dimensional search space is determined by its position and

velocity vectors Suppose that in a d-dimensional optimization

FIGURE 2 Graphical representation of direction indicator

of component x i moving from x a to x b: (a) Scenario 1 and (b) Scenario3

FIGURE 3 Flowchart of the proposed IPG-PSO method for

solving the ORPD problem

problem, the position and velocity vectors of the ith particle

are represented as x i = [x i1, ., x id ] and v i = [v i1, ., v id],

respectively, where i = 1, , np and np is the number of

particles Each particle in the search space is recognized in

the best position by itself, which is called the personal best

Limits of reactive power of shunt VAR compensator (p.u.)

Qmax

ci 5 5 5 5 5 5 5 5 5

Qmin

ci 0 0 0 0 0 0 0 0 0

Limits of reactive power generation (MVAR)

Qmax

Gi 200 100 80 60 50 60

Qmin

Limits of generator bus voltage, load bus voltage, and transformer

tap setting (p.u.)

Vmax

Gi  Vmin

Gi Vmax

Li Vmin

Li Tmax

i Tmin

i

1.10 0.95 1.05 0.95 1.10 0.90

TABLE 2 Limits of control variables

Total real power demand (MW),

P Di 283.4

Total reactive power demand (MVAR),

Q Di 126.2

Reactive power loss (MVAR), Q loss 23.14

Total of real power generation (MW),

P Gi 288.67

Total of reactive power generation (MVAR),

Q Gi 89.09

TABLE 3 Base case power flow solution on the IEEE 30-bus system

or pbest, and the best of the group among pbest is called the global best or gbest Both pbest and gbest can be represented

as pbest i = [pbest i1, ., pbest id ] and gbest = [gbest1, ., gbest d], respectively Thus, the updating velocity and position

of each particle can be given as

ν k+1

i d = ω k × ν k

i d + c1× rand1×pbest i d k − x k

i d



+ c2× rand2×gbest d k − x k

i d



, (20)

x i d k+1= x k

i d + ν k+1

where

ν k+1

i d , x k+1

i d are velocity and position updating of the dth di-mension of the ith particle at iteration k+ 1, respectively;

ν k

i d , x k

i d are current velocity and position updating of the dth dimension of the ith particle at iteration k, respectively;

r and iis a uniformly random number in the range 0 and 1;

c1, c2are cognitive and social acceleration coefficients, respec-tively;

ω k

is an inertia weight factor at iteration k;

pbest k

i d is a personal best position of the dth dimension of the

ith particle at iteration k;

gbest k is a global best position of the dth dimension at iteration

k; and

k is an iteration counter.

3.1 Improved PSO by Chaotic Weight Factor

The chaos theory is very useful for system analysis and predic-tion and also controlling and stabilizing in the system toward

a better global optimum because this theory has the capability

to escape the local optimum To diversify the search space, the chaotic dynamic system is applied to a linearly inertia weight method in the updating velocity process One of the dynamic

systems is employed, that is logistic map, which can be defined

as [24–26]

η k = ϕ × η k−1×1− η k−1

∈ (0, 1) , (22)

Method PSO-TVIW PSO-TVAC SPSO-TVAC PSO-CF PG-PSO SWT-PSO PGSWT-PSO IPG-PSO

TABLE 4 PSO parameters

Minimum P loss 4.8458 4.8449 4.5262 4.5258 4.6425 4.6578 4.7914 4.5256

Average P loss 4.8761 4.8702 4.5564 4.5711 4.7320 4.9413 5.2349 4.5508

Maximum P loss 5.2292 4.9655 4.7716 4.9990 4.7972 5.2521 6.0512 4.9493

Standard

deviation

P loss

0.0562 0.0263 0.0554 0.0815 0.1124 0.1221 0.2131 0.0592

VD 0.9522 0.9408 1.8811 1.8589 0.9321 1.5206 0.6959 1.7854

Lmax 0.1376 0.1376 0.1273 0.1276 0.1378 0.1318 0.1425 0.1223

Average CPU

time (sec)

TABLE 5 Best result comparison of ORPD minimizing real power loss (P loss) by IPG-PSO method on the IEEE 30-bus system

where ϕ is a control parameter, which is equal to 4; η kis a

chaotic parameter at iteration k; and the initial chaotic

param-eter is represented asη(0)∈ (0, 1) − {0.25, 0.50, 0.75}

The inertia weight factor is linearly decreasing to enhance

the exploration and exploitation in the search space A high

weight value can diversify global search, whereas a low value

can intensify local search Hence, the inertia weight factor is

decreasing from maximum to minimum weight factors as

ω k = ωmax−ωmax− ωmin

kmax

To properly control of the inertia weight, a linearly

decreas-ing chaotic inertia weight factor is applied in Eq (23) in lieu

of a constant or linearly decreasing inertia weight factor so as

Maximum

VD

0.5791 0.5796 0.1833 0.4041 0.2593 0.2296 0.5532 0.2518

Standard

deviation

VD

0.1112 0.0153 0.0103 0.0404 0.0222 0.133 0.0656 0.0298

P loss(MW) 5.8452 5.3829 5.7269 5.8258 5.6428 5.7026 5.4886 5.7429

Lmax 0.1481 0.1485 0.1484 0.1485 0.1490 0.1487 0.1474 0.1487

Average CPU

time (sec)

TABLE 6 Best result comparison of ORPD minimizing voltage deviation (VD) by IPG-PSO method on the IEEE 30-bus system

Methods PSO-TVIW PSO-TVAC SPSO-TVAC PSO-CF PG-PSO SWT-PSO PGSWT-PSO IPG-PSO Minimum

Lmax

0.1258 0.1499 0.1271 0.1261 0.1264 0.1488 0.1394 0.1241

Maximum

Lmax

0.1289 0.1544 0.1297 0.1295 0.1313 0.1806 0.1749 0.1298

Standard

deviation

Lmax

0.0008 0.0009 0.0006 0.0008 0.0008 0.0074 0.0081 0.0010

P loss(MW) 4.9186 4.8599 5.2558 5.0041 4.6603 5.0578 5.1142 5.0644

VD 1.9427 1.9174 1.6830 1.9429 1.9940 1.3788 1.30974 1.8987

Average CPU

time (sec)

TABLE 7 Best result comparison of ORPD minimizing voltage stability index (Lmax) by IPG-PSO method on the IEEE 30-bus system

FIGURE 4 Convergence characteristics of PSO methods

min-imizing real power loss on the IEEE 30-bus system

FIGURE 5 Convergence characteristics of PSO methods

min-imizing voltage deviation on the IEEE 30-bus system

FIGURE 6 Convergence characteristics of PSO methods

min-imizing the voltage stability index on the IEEE 30-bus system

to improve capability for global searching and escaping from the local minimum defined as

c ω k = ω k × η k , (24)

where c ω k

is the chaotic weight at iteration k, and ω k

is the

inertia weight factor at iteration k.

As shown in Figure 1, the proposed linearly chaotic weight factor is decreasing and oscillating simultaneously, whereas the linearly decreasing inertia weighting factor decreases incessantly from a maximum weighting factor (ωmax) to a min-imum weighting factor (ωmin)

3.2 Pseudo-gradient Concept

The pseudo-gradient search method is used to determine the search direction for each particle in a population It can provide

a good direction in the search space of a problem without

Methods Real power loss

Voltage deviation

Voltage stability index

NMSFLA [16] 4.6118 (V L=

[0.95, 1.10])

GSA [11] 4.5143 (V L=

[0.94, 1.06])

TABLE 8 Best result comparison of ORPD minimizing P loss , VD,

and Lmaxby IPG-PSO method on the IEEE-30 bus system

requiring the objective function to be differentiable Therefore,

the pseudo-gradient search is suitable for implementation on

the meta-heuristic search methods for solving non-linear and

non-convex problems with multiple minima [27]

For a non-differentiable d-dimension optimization problem

with the objective function f (x), where x = [x1, ., x d], a

pseudo-gradient g p (x) for the objective function is defined as

follows [18]

Suppose that x a = [x a1, ., x ad] is a point in the search

space of the problem, and it moves to another point x b Thus,

Number of generator buses

54

Number of capacitor banks

14 Number of control

variables

77 Total real power demand

(MW),

P Di

4,242 Total reactive power

demand (MVAR),



Q Di

1,438

Real power loss (MW),

P loss

132.863

Reactive power loss

(MVAR), Q loss

783.79

Total of real power generation (MW),



P Gi

4, 374.86

Total of reactive power generation (MVAR),



Q Gi

795.68

TABLE 10 Base case power flow solution on the IEEE 118-bus

system

the exploration of the pseudo-gradient can be divided into two cases for this movement

Case 1: If f (x b)< f (x a ), the direction from x a to x bis defined

as the positive direction The pseudo-gradient at point x b,

g p (x b), is determined by

g p (x b)= [δ(x b1), ,δ(x bd)]T (25)

Limits of reactive power shunt VAR compensator (p.u.)

Qmax

ci 0 14 0 10 10 10 15 12 20 20 10 20 6 6

Qmin

Limits of reactive power generation (MVAR)

Qmax

Gi Qmin

Gi

Limits of generator bus voltage, load bus voltage, and transformer tap setting (p.u.)

Vmax

Gi  Vmin

Gi  Vmax

Li  Vmin

Li  Tmax

i Tmin

i

1.10 0.95 1.05 0.95 1.10 0.90

TABLE 9 Limits of control variables

The direction indicator of position x b,δ(x bi), moving from

x a to x bcan be defined by

δ (x bi)=

⎧

⎨

⎩

1 0

−1

i f x bi > x ai

i f x bi = x ai

i f x bi < x ai

. (26)

Case 2: If f (x b)> f (x a ), the direction from x a to x bis defined

as the negative direction The pseudo-gradient at point x bis

determined by

g p (x b)= 0. (27) For simplicity, the characteristic of the direction indicator

is shown in Table 1 It can be classified into six cases

cor-responding to Eq (25)–(27) The graphical illustration of

Scenarios 1 and 3 is indicated in Figure 2

The pseudo-gradient can also indicate a better direction

similar to the conventional gradient in the search space based

on the two last points If the pseudo-gradient at point x bis not

equal to zero, it implies that a better solution for the objective

function could be found in the next step based on the direction

indicated by the pseudo-gradient at point x b Otherwise, the

search direction at this point should not be changed

3.3 IPG-PSO

In this article, the proposed IPG-PSO uses a linearly decreasing

chaotic inertia weight factor and is guided by the

pseudo-gradient search algorithm To enhance the search direction of

the position, two points are considered: x a and x b In particular,

when the position of a particle needs to be updated, the new

position in the next iteration, x k+ 1, can be represented as

x i d k+1=



x k

i d + δx k+1

i d



×vi d k+1

x k

i d + v k+1

i d

i f g p



x k i d+1

= 0, otherwise.

(28)

From Eq (28), if the pseudo-gradient is not equal to zero, the position of the particle is updated in an appropriate direc-tion to a better soludirec-tion Otherwise, the posidirec-tion is updated by the original one in Eq (21)

3.4 IPG-PSO for ORPD Problem

The IPG-PSO method optimizes regulated generator volt-ages, tap settings of the OLTC, and reactive power injections minimizing fitness function in Steps 1–7 of the detailed pro-cedure given next

Step 1: Initialize position and velocity of swarm Identify the

input data, including gen data, bus data, line data, constraint limits, and PSO parameters (including the number of

par-ticles [np], c1, c2, wmax, wmin, maximum iteration [Itermax], control parameter, and initial chaotic parameter)

To implement this algorithm for solving the ORPD problem, the position and velocity of each particle representing the control variables, such as generator bus voltages, outputs of shunt VAR compensators, and tap settings of on-load tap changing transformers, are randomly generated within their limits

The control variables can be represented as

x i(0)=V1

Gi   V Gi N G , T i1 T N T

i , Q1

ci Q N C ci

T

,

To compute the upper and lower bounds of velocity, each particle is constrained within the minimum and maximum values of position as

νmax

i = LF ×x imax− xmin

i



, (30)

νmin

i = −νmax

where LF is the limit factor of the dynamic range of value

of each dimension, i.e., usually set to 15–20%.

Minimum

P loss

116.8976 124.3335 116.2026 115.6469 116.6075 124.1476 119.4271 115.0605

Average P loss 118.2344 129.7494 117.3553 116.9863 119.3968 129.3710 122.7814 116.4629

Maximum

P loss

126.6222 134.1254 118.1390 119.8378 127.0772 141.6147 125.7621 118.3502

Standard

deviation

P loss

1.6009 2.1560 0.4696 0.8655 2.1070 3.3090 1.2455 0.5280

VD 2.0219 1.4332 1.8587 2.1306 2.2002 1.1459 1.4193 2.1152

Lmax 0.0644 0.0679 0.0650 0.0647 0.0651 0.0666 0.0646 0.0641

Average CPU

time (sec)

TABLE 11 Best result comparison of ORPD minimizing real power loss (P ) by IPG-PSO method on the IEEE 118-bus system