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Electric Power Systems Research 81 (2011) 458464

Contents lists available atScienceDirect

Electric Power Systems Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e p s r

Differential evolution algorithm for optimal reactive power dispatch

A.A. Abou El Ela a, M.A. Abido b, S.R. Spea a,

a Electrical Engineering Department, Faculty of Engineering, Menoufiya University, Egyptb Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Saudi Arabia

a r t i c l e i n f o

Article history:

Received 21 August 2009

Received in revised form 12 August 2010

Accepted 4 October 2010Available online 30 October 2010

Keywords:

Reactive power dispatch

Differential Evolution algorithm

Fuel cost minimization

Voltage profile improvement

Voltage stability enhancement

a b s t r a c t

Reactive power dispatch(RPD)is oneof theimportanttasksin theoperation and control of power system.

This paper presents an efficient and reliable evolutionary-basedapproach to solve the RPD problem. The

proposed approach employs differential evolution (DE) algorithm for optimal settings of RPD control

variables. The proposed approach is examined and tested on the standard IEEE 30-bus test system with

different objectives that reflect power losses minimization, voltage profile improvement, and voltage

stabilityenhancement. Thesimulation resultsof theproposedapproachare comparedto those reported in

theliterature. The results demonstrate thepotential of theproposedapproachand show itseffectiveness

and robustness to solve the RPD problem.

Crown Copyright 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction

The purpose of the reactive power dispatch (RPD) in power sys-

tem is to identify the control variables which minimize the given

objective function while satisfying the unit and system constraints.

This goal is achieved by proper adjustment of reactive power vari-

ables like generator voltage magnitudes, switchable VAR sources

and transformer tap setting[1].

In thepast twodecades,the problem of RPDfor improvingecon-

omy and security of power system operation has received much

attention. The main objective of optimal reactive power control is

to improve the voltage profile and minimizing system real power

losses via redistribution of reactive power in the system. In addi-

tion, the voltage stability can be enhanced by reallocating reactive

power generations. Therefore, the problem of the RPD can be opti-

mized to enhance the voltage stability, improve voltage profile and

minimize the system losses as well[24].

To solve the RPD problem, a number of conventional opti-

mization techniques [5,6] have been proposed. These includethe Gradient method, Non-linear Programming (NLP), Quadratic

Programming (QP), Linear programming (LP) and Interior point

method. Though these techniques have been successfully applied

for solving the reactive power dispatch problem, still some dif-

ficulties are associated with them. One of the difficulties is the

multimodal characteristic of the problems to be handled. Also, due

to the non-differential, non-linearity and non-convex nature of the

Corresponding author.

E-mail address:shi [email protected](S.R. Spea).

RPD problem, majority of the techniques converge to a local opti-

mum. Recently, Evolutionary Computation techniques like Genetic

Algorithm (GA)[7],Evolutionary Programming (EP)[8]and Evolu-

tionary Strategy [9] havebeen applied to solve theoptimal dispatch

problem. In this paper,a new evolutionarycomputation technique,

called Differential Evolution (DE) algorithm is used to solve RPD

problem.

Recently, differential evolution (DE) algorithm has been pro-

posed and introduced [1013]. The algorithm is inspired by

biological and sociological motivations and can take care of opti-

mality on rough, discontinuous and multi-modal surfaces. The DE

has threemain advantages: it can findnear optimal solution regard-

less the initial parameter values, its convergence is fast and it uses

few numberof control parameters. In addition, DE is simple in cod-

ing and easy to use. It can handle integer and discrete optimization

[1013].

The performance of DE algorithm was compared to that of dif-

ferent heuristic techniques. It is found that, the convergence speed

of DE is significantly better than that of GAs [12].In[14],the per-formance of DE was compared to PSO and evolutionary algorithms

(EAs). The comparison is performed on a suite of 34 widely used

benchmark problems. It is found that, DE is the best perform-

ing algorithm as it finds the lowest fitness value for most of the

problems considered in that study. Also, DE is robust; it is able to

reproduce the same results consistently over many trials, whereas

the performance of PSO is far more dependent on the randomized

initialization of the individuals[14].In addition, the DE algorithm

has been used to solve high-dimensional function optimization(up

to 1000 dimensions)[15].It is found that, it has superior perfor-

mance on a set of widely used benchmark functions. Therefore, the

0378-7796/$ see front matter. Crown Copyright 2010 Published by Elsevier B.V. All rights reserved.

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A.A.A.E. Ela et al. / Electric Power Systems Research 81 (2011) 458464 459

DE algorithm seems to be a promising approach for engineering

optimization problems.It has successfully beenappliedand studied

to many artificial and real optimization problems[1620].

In[2123], the DE algorithm is used as an optimization tool for

the reactive power optimization with the propose of minimizing

the system power losses while maintaining the dependant vari-

ables including voltages of PQ-buses and reactive power outputs of

generators, within limits. In[24],the DE algorithm is used to solve

reactive power dispatch for voltage stability enhancement.

Inthis paper,a novel DE-based approach is proposedto solve the

RPD problem. The problem is formulated as a nonlinear optimiza-

tion problem with equality and inequalityconstraints. In this study,

different objectives are considered such as minimizing the power

losses, improving the voltage profile, and enhancing power sys-

tem voltage stability. The proposed approach has been examined

and tested on the standard IEEE 30-bus test system. The poten-

tial and effectiveness of the proposed approach are demonstrated.

Additionally, the results are compared to those reported in the

literature.

2. Problem formulation

The objective of RPD is to identify the reactive power control

variables, which minimizes the objective functions. This is mathe-matically stated as follows:

2.1. Problem objectives

In this study, the following objectives are considered:

2.1.1. Minimization of system power losses

The minimization of system real power losses Ploss(MW) can

be calculated as follows:

f1 = Ploss =

nlk=1

gk[V2i + V

2j 2ViVjcos(i j)] (1)

where nl is the number of transmission lines;gkis the conductanceof the kth line; Viand Vjare the voltage magnitude at the end buses

i and j of the kth line, respectively, and i and j are the voltage

phase angle at the end buses i andj.

2.1.2. Voltage profile improvement

Bus voltage is one of the most important security and service

quality indices. Improving voltage profile can be obtained by min-

imizing the load bus voltage deviations from 1.0 per unit. The

objective function can be expressed as:

f2 =iNL

Vi 1.0 (2)whereNLis the number of load buses.

2.1.3. Voltage stability enhancement

It is very important to maintain constantly acceptable bus volt-

age at each bus under normal operating conditions, after load

increase, following systemconfiguration changes, or when the sys-

temis being subjectedto a disturbance. The non-optimized control

variables may lead to progressive and uncontrollable drop in volt-

age resulting in an eventual widespread voltage collapse.

Enhancing voltage stability can be achieved through minimiz-

ing the voltage stability indicator L-index values at every bus of the

system and consequently the global power system L-index [25].

L-index gives a scalar number to each load bus. This index uses

information on a normalpower flow. L-index is in the range of zero

(no load case) andone (voltage collapse).Details ofL-index calcula-

tion andderivation aregivenin Appendix A. In order to enhance the

voltage stability and move the system far from the voltage collapse

point, the following objective function can be used:

f3 = Lmax (3)

2.2. System constraints

2.2.1. Equality constraints

These constraints represent load flow equations:

PGi PDi Vi

NBj=1

Vj[Gijcos(i j) + Bijsin(i j)] = 0 (4)

QGi QDi Vi

NBj=1

Vj[Gijsin(i j) Bijcos(i j)] = 0 (5)

where i = 1,. . .,NB; NB is the number of buses, PG is theactive power

generated, QGis the reactive power generated, PDis the load active

power,QD is the load reactive power, G ij and B ij are the transfer

conductance and susceptancebetween bus i andbusj, respectively.

2.2.2. Inequality constraintsThese constraints include:

1. Generator constraints: generator voltages, and reactive power

outputs arerestricted by their lower andupperlimits as follows:

VminGi VGi VmaxGi

, i = 1, . . . , N G (6)

QminGi QGi QmaxGi , i = 1, . . . , N G (7)

2. Transformer constraints: transformer tap settings are bounded

as follows:

Tmini Ti Tmaxi

, i = 1, . . . , N T (8)

3. Shunt VAR constraints: shunt VAR compensations are restricted

by their limits as follows:

Qminci Qci Qmaxci , i = 1, . . . , N C (9)

Security constraints: these include theconstraintsof voltages at

load buses and transmission line loadings as follows:

VminLi VLi V

maxLi

, i = 1, . . . , N L (10)

Sli Smaxli

, i = 1, . . . , n l (11)

3. Differential evolution algorithm

3.1. Overview

In 1995, Storn and Price proposed a new floating point encoded

evolutionary algorithm for global optimization and named it dif-

ferential evolution (DE) algorithm owing to a special kind of

differential operator, which they invoked to create new off-spring

from parent chromosomes instead of classical crossover or muta-

tion[10].

Similar to GAs, DE algorithm is a population based algorithm

that uses crossover, mutation and selection operators. The main

differences between the genetic algorithm and DE algorithm are

the selection process and the mutation scheme that makes DE self

adaptive.In DE,all solutionshave thesame chanceof being selected

as parents. DE employs a greedy selection process that is the best

new solution andits parentwins thecompetition providing signifi-

cantadvantage of converging performance overgenetic algorithms.

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Initialization of Chromosomes

Mutation Differential Operator

Crossover

Selection

Fig. 1. DE cycle of stages.

3.2. DE computational flow

DE algorithm is a population based algorithm using three oper-

ators; crossover, mutation and selection. Several optimization

parameters must also be tuned. These parameters have joined

together under the common name control parameters. In fact, there

are only three real control parameters in the algorithm, which are

differentiation (or mutation) constant F, crossover constant CR, and

size of populationNP. The rest of the parameters are dimension ofproblemDthat scales the difficulty of the optimization task; max-

imum number of generations (or iterations) GEN, which may serve

as a stopping condition; and low and high boundary constraints of

variables that limit the feasible area[10,11].

The proper setting of NP is largely dependent on the size of

the problem. Storn and Price [10] remarked that for real-world

engineering problems with D control variables, NP= 20D will prob-

ably be more than adequate, NPas small as 5D is often possible,

although optimal solutions using NP< 2Dshould not be expected.

In [13], Storn and Price set the size of population less than the

recommended NP= 10D in many of their test tasks. In [14], it is

recommended using ofNP4D. In[15], NP= 5D is a good choice

for a first try, and then increase or decrease it by discretion. So, as

a rough principle, several tries before solving the problem may be

sufficient to choose the suitable number of the individuals.

The DE algorithm works through a simple cycle of stages, pre-

sented inFig. 1.

These stages can be cleared as follow:

3.2.1. Initialization

At the very beginning of a DE run, problem independent vari-

ables are initialized in their feasible numerical range. Therefore,

if the jth variable of the given problem has its lower and upper

bound as xLj

and xuj

, respectively, then the jth component of the

ith population members may be initialized as,

xi,j(0) = xL

j + rand(0, 1) (xuj x

Lj ) (12)

where rand(0,1) is a uniformly distributed random numberbetween 0 and 1.

3.2.2. Mutation

In each generation to change each population member Xi(t), a

donor vector vi(t) is created. It is the method of creating this donor

vector, which demarcates between the various DE schemes. How-

ever, in this paper, one such specific mutation strategy known as

DE/rand/1 is discussed.

To create a donor vector vi(t) for each ith member, three param-

eter vectorsxr1,xr2 and xr3are chosen randomly from the current

population and not coinciding with the currentxi. Next, a scalar

numberFscales the difference of any two of the three vectors and

the scaled difference is added to the third one whence the donor

vectorv

i(t) is obtained. The usual choice for Fis a number between

0.4 and 1.0. So, the process for the jth component of each vector

can be expressed as,

vi,j(t+ 1) = xr1,j(t) + F (xr2,j(t) xr3,j(t)) (13)

3.2.3. Crossover

To increase the diversity of the population, crossover operator

is carried out in which the donor vector exchanges its components

with those of the current member Xi(t).

Two types of crossover schemes can be used with DE technique.These are exponential crossover and binomial crossover. Although

the exponential crossover was proposed in the original work of

Storn and Price[10], the binomial variant was much more used in

recent applications[14].

In exponential type, the crossover is performed on the D vari-

ables in one loop as far as it is within the CR bound. The first time

a randomly picked number between 0 and 1 goes beyond the CR

value, no crossover is performed and the remaining variables are

left intact. In binomial type, the crossover is performed on all D

variables as far as a randomly picked number between 0 and 1 is

within theCR value. So for high values ofCR, the exponential and

binomial crossovers yield similar results.

Moreover, in the case of exponential crossover one has to be

aware of the fact that there is a small range ofCR values (typically[0.9, 1]) to which the DE is sensitive. This could explain the rule of

thumb derived for the original variant of DE. On the otherhand, for

the same value ofCR, the exponential variant needs a larger value

for thescaling parameterFin orderto avoidpremature convergence

[26].

In this paper, binomial crossover scheme is used which is per-

formed on allDvariables and can be expressed as:

ui,j(t) =

vi,j(t) if rand(0, 1)< CR

xi,j(t) else (14)

ui,j(t) represents the child that will compete with the parent xi,j(t).

3.2.4. Selection

To keep the population size constant over subsequent genera-

tions,the selectionprocess is carried outto determine which oneof

the child and the parent will survive in the next generation, i.e., at

time t = t + 1. DE actually involves the Survival of the fittest princi-

ple in its selection process. The selection process can be expressed

as,

Xi(t+ 1) =

Ui(t) if f( Ui(t)) f(Xi(t))Xi(t) if f(Xi(t))< f( U(t))

(15)

where,f() is thefunctionto be minimized. From Eq.(10) we noticed

that:

Ifui(t) yields a better value of the fitness function, it replaces its

target Xi(t) in the next generation. Otherwise, Xi(t) is retained in the population.

Hence, the population either gets better in terms of the fitness

function or remains constant but never deteriorates.

3.3. DE-based approach implementation

The proposed DE-based approach has been developed and

implemented using the MATLAB software. Several runs have been

done with different values of DE key parameters such as differ-

entiation (or mutation) constant F, crossover constant CR, size of

population NP, and maximum number of generations GENwhich is

used here as a stopping criteria to find the optimal DE key param-

eters. In this paper, the following values of DE key parameters are

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selected for the optimization of power losses and voltage stability

enhancement:

F= 0.2; CR = 0.6; NP= 150; GEN= 500

and DE key parameters for the optimization of voltage deviations

are selected as:

F= 0.2; CR = 0.6; NP= 50; GEN= 500

The first step in the algorithm is creating an initial population.All the independent variables which include generator voltages,

transformer tap settings and shunt VAR compensations have to be

generated according to Eq.(12),where each independent parame-

ter of each individual in the population is assigned a value inside

itsgiven feasible region. This creates parentvectors of independent

variables for the first generation.

After, finding the independent variables, dependent variables

will be found from a load flow solution. These dependent variables

include generatorsreactivepower, voltages at loadbuses andtrans-

mission line loadings. It should be mentioned that, the real power

settings of the generators are taken from[4].

4. Results and discussion

The proposed DE-based algorithm has been tested on the stan-

dard IEEE 6-generator 30-bus test system shown in Fig. 2. The

systemdatais givenin Appendix B [27]. This system has 19-control

variable as follows: 6-generator voltage magnitude, 4-tap trans-

former setting and 9-switchable VAR.

To demonstrate the effectiveness of the proposed algorithm,

three different cases have been considered as follows:

Case1: Minimization of system power losses.

Case2: Improvement of voltage profile.

Case3: Enhancement of voltage stability.

4.1. Case1 (minimization of system power losses)

In the first case, the proposed algorithm is run with minimiza-

tion of real power losses as the objective function. As mentioned

above, the real power settings of the generators are taken from [2].

The convergence characteristic of the algorithm is shown inFig. 3.

The algorithm reaches a minimum loss of 4.5550MW. The optimal

29

30

27 28

2526

2423

191815

2017

21

221614

10

6

911

1

2 5

7

84

1213

3

Fig. 2. Single line diagram of IEEE 30-bus test system.

values of the control variables are given in the second column of

Table 1.The minimum loss obtained by the proposed algorithm is

compared with the results reported in[24]using the evolution-

ary computation techniques for the same test system. The results

of the comparison are given inTable 2.From the comparison, the

proposed algorithm gives the minimum losses which demonstrate

the effectiveness of the proposed algorithm.

4.2. Case2 (improvement of voltage profile)

In the second case, the proposed DE-based approach is applied

for improvement of voltage profile. The convergence characteristic

Table 1

Optimal settings of control variables for different cases.

Case1: minimization of power losses Case2: voltage profile improvement Case3: voltage stability enhancement

V1 1.1000 1.0100 1.0993

V2 1.0931 0.9918 1.0967

V5 1.0736 1.0179 1.0990

V8 1.0756 1.0183 1.0346

V11 1.1000 1.0114 1.0993V13 1.1000 1.0282 0.9517

T11 1.0465 1.0265 0.9038

T12 0.9097 0.9038 0.9029

T15 0.9867 1.0114 0.9002

T36 0.9689 0.9635 0.9360

Qc10 5.0000 4.9420 0.6854

Qc12 5.0000 1.0885 4.7163

Qc15 5.0000 4.9985 4.4931

Qc17 5.0000 0.2393 4.5100

Qc20 4.4060 4.9958 4.4766

Qc21 5.0000 4.9075 4.6075

Qc23 2.8004 4.9863 3.8806

Qc24 5.0000 4.9663 4.2854

Qc29 2.5979 2.2325 3.2541

Power losses (MW) 4.5550 6.4755 7.0733

Voltage deviations 1.9589 0.0911 1.4191

Lmax 0.5513 0.5734 0.1246

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0 50 100 150 200 250 300 350 400 450 5004.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

Generations

Ploss(MW)

Fig. 3. Power losses variations of Case1.

Table 2

Comparison of power losses for different methods.

Method Ploss(MW)

Strength pareto evolutionary algorithm[2] 5.1170

Genetic algorithm[3] 4.5800

Genetic algorithm-based approach[4] 4.6501

Proposed algorithm 4.5550

of the algorithm for this case is shown inFig. 4.The optimal values

of the control variable settings obtained in this case are given in

the third column ofTable 1.In this case, the voltage deviations are

reduced from 1.1606 in theinitialstate to0.0911witha reductionof

92.15%. The comparison with the results reported in [1]is given in

Table 3where 79.11% reduction is achieved. From the comparison,

the proposed algorithmgives thebest results forvoltage deviations

which demonstrate the effectiveness of the proposed algorithm.

0 50 100 150 200 250 300 350 400 450 5000.05

0.1

0.15

0.2

0.25

0.3

Generations

VoltageDeviations

Fig. 4. Voltage deviations variations of Case3.

Table 3

Comparison of voltage deviations for different methods.

Method Voltage deviations

Particle swarm optimization[1] 0.2424


0 50 100 150 200 250 300 350 400 450 5000.124

0.125

0.126

0.127

0.128

0.129

0.13

Generations

L

max

Fig. 5. Lmaxvariations of Case3.

Table 4

Comparison ofLmaxvalue for different methods.

Method Lmax

Strength pareto evolutionary algorithm[2] 0.1397


4.3. Case3 (enhancement of voltage stability)

In the third case, the proposed DE-based approach is applied

for enhancement of voltage stability as the objective. The conver-

gence characteristic of the algorithmfor this case is shown in Fig. 5.

The optimal values of the control variable settings obtained in this

case are given in the fourth column ofTable 1. In this case, the

maximumL-index of the system has been reduced from 0.2144

in the initial state to 0.1246. Hence, it is observed that there is

an increase in performance of the system. Thus it results in theenhancement of voltage stability level of the system. The results of

the comparison with the results reported in the literature are given

inTable 4.From the comparison, the proposed algorithm gives the

best results for Lmax which demonstrates the effectiveness of the

proposed algorithm.

5. Conclusions

In this paper, a differential evolution (DE) optimization algo-

rithm has been proposed, developed, and successfully applied to

solve reactive power dispatch (RPD) problem. TheRPD problem has

been formulated as a constrained optimization problem wheresev-

eral objective functions have been considered to minimize power

losses, to improve the voltage profile, and to enhance the voltage

stability. The proposed approach has been tested and examined

on the standard IEEE 30-bus test system. The simulation results

demonstrate the effectiveness and robustness of the proposedalgo-

rithm to solve RPD problem. Moreover, the results of the proposed

DE algorithm have been compared to those reported in the litera-

ture.The comparison confirms the effectiveness and the superiority

of the proposed DE approach over the classical and heuristic tech-

niques in terms of solution quality.

Acknowledgement

Dr. M.A. Abido would like to acknowledge the support of King

Fahd University of Petroleum & Minerals.

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Appendix A. L-index calculation

For voltage stabilityevaluation, an indicator L-indexis used. The

indicator value varies in the range between 0 (the no load case)

and 1 which corresponds to voltage collapse. The indicator uses

bus voltage and network information provided by the load flow

program.

Table A1Load data.

Bus no. Load Bus no. Load

P(p.u.) Q(p.u.) P(p.u.) Q(p.u.)

1 0.000 0.000 16 0.035 0.018

2 0.217 0.127 17 0.090 0.058

3 0.024 0.012 18 0.032 0.009

4 0.076 0.016 19 0.095 0.034

5 0.942 0.190 20 0.022 0.007

6 0.000 0.000 21 0.175 0.112

7 0.228 0.109 22 0.000 0.000

8 0.300 0.300 23 0.032 0.016

9 0.000 0.000 24 0.087 0.067

10 0.058 0.020 25 0.000 0.000

11 0.000 0.000 26 0.035 0.023

12 0.112 0.075 27 0.000 0.000

13 0.000 0.000 28 0.000 0.000

14 0.062 0.016 29 0.024 0.009

15 0.082 0.025 30 0.106 0.019

Table A2

Line data.

Line no. From bus To bus Line impedance

R(p.u.) X(p.u.)

1 1 2 0.0192 0.0575

2 1 3 0.0452 0.1852

3 2 4 0.0570 0.1737

4 3 4 0.0132 0.0379

5 2 5 0.0472 0.1983

6 2 6 0.0581 0.1763

7 4 6 0.0119 0.04148 5 7 0.0460 0.1160

9 6 7 0.0267 0.0820

10 6 8 0.0120 0.0420

11 6 9 0.0000 0.2080

12 6 10 0.0000 0.5560

13 9 11 0.0000 0.2080

14 9 10 0.0000 0.1100

15 4 12 0.0000 0.2560

16 12 13 0.0000 0.1400

17 12 14 0.1231 0.2559

18 12 15 0.0662 0.1304

19 12 16 0.0945 0.1987

20 14 15 0.2210 0.1997

21 16 17 0.0824 0.1932

22 15 18 0.1070 0.2185

23 18 19 0.0639 0.1292

24 19 20 0.0340 0.0680

25 10 20 0.0936 0.2090

26 10 17 0.0324 0.0845

27 10 21 0.0348 0.0749

28 10 22 0.0727 0.1499

29 21 22 0.0116 0.0236

30 15 23 0.1000 0.2020

31 22 24 0.1150 0.1790

32 23 24 0.1320 0.2700

33 24 25 0.1885 0.3292

34 25 26 0.2544 0.3800

35 25 27 0.1093 0.2087

36 28 27 0.0000 0.3960

37 27 29 0.2198 0.4153

38 27 30 0.3202 0.6027

39 29 30 0.2399 0.4533

40 8 28 0.6360 0.2000

41 6 28 0.0169 0.0599

Table A3

Generator data.

Bus no. Cost coefficients

a b c

1 0.00 2.00 0.00375

2 0.00 1.75 0.01750

5 0.00 1.00 0.06250

8 0.00 3.25 0.00834

11 0.00 3.00 0.02500

13 0.00 3.00 0.02500

For multi-node system:

Ibus= Ybus Vbus (A.1)

By segregating the load buses (PQ) from generator buses (PV),

Eq.(A.1)can be rewritten asILIG

=

Y1 Y2Y3 Y4

VLVG

(A.2)

VLIG

=

H1 H2H H4

ILVG

(A.3)

where VL, IL is the voltages and currents for PQ buses; VG, IG isthe voltages and currents for PV buses; H1, H2, H3, and H4 is the

submatrices generated fromYbuspartial inversion.

Let

Vok =

H2ki Vi (A.4)

whereNGis the number of generators

H2 = Y1 Y2 (A.5)

Lk =

1 + VokVk

(A.6)whereLkis theL-index voltage stability indicator for bus k.

Stability requires thatLk < 1 and must not be violated on a con-

tinuous basis. Hence a global system indicator L describing the

Table A4

The minimum and maximum limits for the control variables along with the initial

settings.

Min. Max. Initial

P1 50 200 99.24

P2 20 80 80.0

P5 15 50 50.0

P8 10 35 20.0

P11 10 30 20.0

P13 12 40 20.0

V1 0.95 1.1 1.05

V2 0.95 1.1 1.04

V5 0.95 1.1 1.01

V8 0.95 1.1 1.01

V11

0.95 1.1 1.05

V13 0.95 1.1 1.05

T11 0.9 1.1 1.078

T12 0.9 1.1 1.069

T15 0.9 1.1 1.032

T36 0.9 1.1 1.068

Qc10 0.0 5.0 0.0

Qc12 0.0 5.0 0.0

Qc15 0.0 5.0 0.0

Qc17 0.0 5.0 0.0

Qc20 0.0 5.0 0.0

Qc21 0.0 5.0 0.0

Qc23 0.0 5.0 0.0

Qc24 0.0 5.0 0.0

Qc29 0.0 5.0 0.0

Power losses (MW) 5.842

Voltage deviations 1.1606

Lmax 0.2144

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stabilityof thecompletesystem is L = Lmax{Lk}, where{Lk} contains

Lindices of all load buses.

In practice Lmax must be lower than a threshold value. The

predetermined threshold value is specified at the planning stage

depending on the system configuration and on the utility policy

regarding the quality of service and the level of system decided

allowable margin.

The objective is to minimizeLmax, that is,

Lmax= max

Lk

, k = 1, . . . , N L (A.7)

Appendix B. System data

Data for IEEE 30-bus test system (100 MVA base) are given in

Tables A1A4.

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