Optimal Power Flow Ieee

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002 229

Optimal Power Flow by Enhanced Genetic AlgorithmAnastasios G. Bakirtzis , Senior Member, IEEE, Pandel N. Biskas , Student Member, IEEE,

Christoforos E. Zoumas, Student Member, IEEE, and Vasilios Petridis, Member, IEEE

AbstractThis paper presents an enhanced genetic algorithm(EGA) for the solution of the optimal power flow (OPF) withboth continuous and discrete control variables. The continuouscontrol variables modeled are unit active power outputs andgenerator-bus voltage magnitudes, while the discrete ones aretransformer-tap settings and switchable shunt devices. A numberof functional operating constraints, such as branch flow limits,load bus voltage magnitude limits, and generator reactive capa-bilities, are included as penalties in the GA fitness function (FF).Advanced and problem-specific operators are introduced in orderto enhance the algorithms efficiency and accuracy. Numericalresults on two test systems are presented and compared withresults of other approaches.

Index TermsGenetic algorithms (GAs), optimal power flow(OPF).

NOMENCLATURE

Bus voltage angle vector.

Load (PQ) bus voltage magnitude vector.

Unit active power output vector.

Generation (PV) bus voltage magnitude vector.

Transformer tap settings vector.

Bus shunt admittance vector.

System state vector.

System control vector.

A hat above vectors and denotes that the entry corre-

sponding to the slack bus is missing. For simplicity of notation,

it is assumed that there is only one generating unit connected on

a bus. This assumption is relaxed in SectionV.

I. INTRODUCTION

SINCE its introduction as network constrained economic

dispatch by Carpentier [1] and its definition as optimal

power flow (OPF) by Dommel and Tinney [2], the OPF

problem has been the subject of intensive research.

The OPF optimizes a power system operating objective func-

tion (such as theoperating cost of thermal resources) while satis-

fying a set of system operating constraints, including constraints

dictated by the electric network. OPF has been widely used inpower system operation and planning [3]. After the electricity

sector restructuring, OPF has been used to assess the spatial

variation of electricity prices and as a congestion management

and pricing tool [4].

In its most general formulation, the OPF is a nonlinear,

nonconvex, large-scale, static optimization problem with both

Manuscript received October 9, 2000; revised August 20, 2001.Theauthors arewith theDepartment of Electrical Engineering,Aristotle Uni-

versity of Thessaloniki, Thessaloniki 54006, Greece.Publisher Item Identifier S 0885-8950(02)03812-9.

continuous and discrete control variables. Even in the absence

of nonconvex unit operating cost functions, unit prohibited op-

erating zones, and discrete control variables, the OPF problem

is nonconvex due to the existence of the nonlinear (AC) power

flow equality constraints. The presence of discrete control

variables, such as switchable shunt devices, transformer tap

positions, and phase shifters, further complicates the problem

solution.

The literature on OPF is vast, and [5] presents the major con-

tributions in this area. Mathematical programming approaches,

such as nonlinear programming (NLP) [6][9], quadratic

programming (QP) [10], [11], and linear programming (LP)

[12][14], have been used for the solution of the OPF problem.

Some methods, instead of solving the original problem, solve

the problems KarushKuhnTucker (KKT) optimality con-

ditions. For equality-constrained optimization problems, the

KKT conditions are a set of nonlinear equations, which can be

solved using a Newton-type algorithm. In Newton OPF [ 15],

the inequality constraints are added as quadratic penalty terms

to the problem objective, multiplied by appropriate penalty

multipliers. Interior point (IP) methods [16][18], convert

the inequality constraints to equalities by the introduction of

nonnegative slack variables. A logarithmic barrier function

of the slack variables is then added to the objective function,

multiplied by a barrier parameter, which is gradually reduced tozero during the solution process. The unlimited point algorithm

[19] uses a transformation of the slack and dual variables of the

inequality constraints which converts the OPF problem KKT

conditions to a set of nonlinear equations, thus avoiding the

heuristic rules for barrier parameter reduction required by IP

methods.

OPF programs based on mathematical programming ap-

proaches are used daily to solve very large OPF problems.

However, they are not guaranteed to converge to the global

optimum of the general nonconvex OPF problem, although

there exists some empirical evidence on the uniqueness of

the OPF solution within the domain of interest [20]. To avoid

the prohibitive computational requirements of mixed-integerprogramming, discrete control variables are initially treated

as continuous, and post-processing discretization logic is

subsequently applied [21], [22]. Whereas the effects of dis-

cretization on load tap changing transformers are small and

usually negligible [20], the rounding of switchable shunt

devices may lead to voltage infeasibility, especially when the

discrete VAR steps are large, and requires special logic [22].

The handling of nonconvex OPF objective functions, as well as

the unit prohibited operating zones, also present problems to

mathematical programming OPF approaches.

0885-8950/02$17.00 2002 IEEE
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230 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002

Recent attempts to overcome the limitations of the mathemat-

ical programming approaches include the application of simu-

lated annealing-type methods [23], [24], and genetic algorithms

(GAs) [25], [26].

In [25], a simple genetic algorithm (SGA) is used for OPF so-

lution. The control variables modeled are generator active power

outputs andvoltages, shuntdevices,and transformertaps. Branch

flow, reactive generation, and voltage magnitude constraints aretreated as quadratic penalty terms in the GA fitnessfunction(FF).

TokeeptheGAchromosomesizesmall,onlya4-bitchromosome

area is used for the encoding of each control variable. A sequen-

tial GA solution scheme is employed to achieve acceptable con-

trol variable resolution. Test results on the IEEE 30-bus system,

comprising 25 control variables, are presented.

In [26], a GA is used to solve the optimal power dispatch

problem for a multinode auction market. The GA maximizes the

total participants welfare, subject to network flow and transport

limitation constraints. The nodal real and reactive power injec-

tions that clear the market are selected as the problem control

variables. A GA with two advanced operators, namely, elitism

and hill climbing, is used. A 10-bit chromosome area is devotedto each control variable. Test results on a 17-node, 34-control

variable system are presented.

The GA-OPF approaches overcome the limitations of

the conventional approaches in the modeling of nonconvex

cost functions, discrete control variables, and prohibited unit

operating zones. However, they do not scale easily to larger

problems, since the solution deteriorates with the increase of

the chromosome length, i.e., the number of control variables.

Thus, the test results in the existing GA-OPF literature are

limited to very small problems.

This paper presents an enhanced genetic algorithm (EGA) for

the solution of the OPF. The control variables and constraints

included in the OPF and the penalty method treatment of the

functional operating constraints are similar to the ones in [25]

with the following improvements: switchable shunt devices and

transformer taps are modeled as discrete control variables. Vari-

able binary string length is used for different types of control

variables, so as to achieve the desired resolution for each type

of control variable, without unnecessarily increasing the size of

the GA chromosome. In addition to the basic genetic operators

of the SGA used in [25] and the advanced ones used in [26],

problem-specific operators, inspired by the nature of the OPF

problem, have been incorporated in our EGA. With the incor-

poration of the problem-specific operators, the GA can solve

larger OPF problems. Test results on systems with up to 242buses and 500 control variables demonstrate the improvement

achieved with the aid of problem-specific operators.

II. OPTIMAL POWER FLOW PROBLEM FORMULATION

The OPF problem can be formulated as a mathematical opti-

mization problem as follows:

Min (1)

S.t. (2)

(3)

(4)

where

(5)

(6)

The equality constraints (2) are the nonlinear power flow

equations. The inequality constraints (3) are the functional op-erating constraints, such as

branch flow limits (MVA, MW or A);

load bus voltage magnitude limits;

generator reactive capabilities;

slack bus active power output limits.

Constraints (4) define the feasibility region of the problem

control variables such as

unit active power output limits;

generation bus voltage magnitude limits;

transformer-tap setting limits (discrete values);

bus shunt admittance limits (continuous or discrete

control).

III. GENETIC ALGORITHMS

GAs are general purpose optimization algorithms based on

the mechanics of natural selection and genetics. They operate

on string structures (chromosomes), typically a concatenated list

of binary digits representing a coding of the control parameters

(phenotype) of a given problem. Chromosomes themselves are

composed of genes. The real value of a control parameter, en-

coded in a gene, is called an allele [27].

GAs are an attractive alternative to other optimization

methods because of their robustness. There are three major

differences between GAs and conventional optimization algo-

rithms. First, GAs operate on the encoded string of the problem

parameters rather than the actual parameters of the problem.

Each string can be thought of as a chromosome that completely

describes one candidate solution to the problem. Second, GAs

use a population of points rather than a single point in their

search. This allows the GA to explore several areas of the

search space simultaneously, reducing the probability of finding

local optima. Third, GAs do not require any prior knowledge,

space limitations, or special properties of the function to be

optimized, such as smoothness, convexity, unimodality, or

existence of derivatives. They only require the evaluation of

the so-called fitness function (FF) to assign a quality value to

every solution produced.Assuming an initial random population produced and evalu-ated, genetic evolution takes place by means of three basic ge-

netic operators:

1) parent selection;

2) crossover;

3) mutation.

Parent selection is a simple procedure whereby two chromo-

somes are selected from the parent population based on their fit-

ness value. Solutions with high fitness values have a high prob-

ability of contributing new offspring to the next generation. The

selection rule used in our approach is a simple roulette-wheel

selection [27].
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BAKIRTZIS et al.: OPTIMAL POWER FLOW BY ENHANCED GENETIC ALGORITHM 231

Fig. 1. Simple genetic algorithm (SGA).

Crossover is an extremely important operator for the GA.

It is responsible for the structure recombination (information

exchange between mating chromosomes) and the conver-

gence speed of the GA and is usually applied with high

probability (0.60.9). The chromosomes of the two parents

selected are combined to form new chromosomes that in-

herit segments of information stored in parent chromosomes.

Until now, many crossover schemes, such as single point,

multipoint, or uniform crossover have been proposed in the

literature. Uniform crossover [28] has been used in our

implementation.

While crossover is the main genetic operator exploiting theinformation included in the current generation, it does not pro-

duce new information.

Mutation is the operator responsible for the injection of new

information. With a small probability, random bits of the off-

spring chromosomes flip from 0 to 1 and vice versa and give

new characteristics that do not exist in the parent population

[27]. In our approach, the mutation operator is applied with a

relatively small probability (0.0001-0.001) to every bit of the

chromosome.

The FF evaluation and genetic evolution take part in an iter-

ative procedure, which ends when a maximum number of gen-

erations is reached, as shown in Fig. 1.When applying GAs to solve a particular optimization

problem (OPF in our case), two main issues must be

addressed:

1) the encoding, i.e., how the problem physical decision

variables are translated to a GA chromosome and its

inverse operator, decoding;

2) the definition of the FF to be maximized by the GA

(the GA FF is formed by an appropriate transformation

of the initial problem objective function augmented by

penalty terms that penalize the violation of the problem

constraints [29]).

Fig. 2. GA chromosome structure.

IV. GENETIC ALGORITHM SOLUTION TO

OPTIMAL POWER FLOW

A. Encoding

The chromosome is formed as shown in Fig. 2. There are

four chromosome regions (one for each set of control vari-

ables), namely, 1) ; 2) ; 3) ; and 4) . Encoding is

performed using different gene-lengths for each set of control

variables, depending on the desired accuracy. The decoding of

a chromosome to the problem physical variables is performed

as follows:

1) continuous controls taking values in the interval

(7)

2) discrete controls taking values

with

and

(8)

where is the decimal number to which the binary

number in a gene is decoded and is the gene length

(number of bits) used for encoding control variable .

B. Fitness Function (FF)

GAs are usually designed so as to maximize the FF, which

is a measure of the quality of each candidate solution. The ob-

jective of the OPF problem is to minimize the total operating

cost (1).

Therefore, a transformation is needed to convert the cost

objective of the OPF problem to an appropriate FF to be max-

imized by the GA. The OPF functional operating constraints

(3) are included in the GA solution by augmenting the GA

FF by appropriate penalty terms for each violated functional

constraint. Constraints on the control variables (4) are au-

tomatically satisfied by the selected GA encoding/decoding

scheme (7) and (8).

Therefore, the GA FF is formed as follows:

(9)

(10)
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where

fitness function;

constant;

fuel cost function of unit (in our case, a quadratic

function);

weighting factor of functional operating constraint

;

penalty function for functional operating constraint

;

violation of th functional operating constraint, if

positive;

Heaviside (step) function;

number of units;

number of functional operating constraints.

Given a candidate solution to the problem, represented by a

chromosome, the FF is computed as follows.

Step 1) Decode the chromosome to determine the actual

control variables, , using (7) and (8). The com-

puted control vector satisfies, by design, constraints(4).

Step 2) Solve the power flow (2) to compute the state vector,

.

Step 3) Determine the violated functional constraints (3) and

compute associated penalty functions (10).

Step 4) Compute the FF using (9).

In Step 2, a simple fast decoupled load flow (FDLF) [30]

is used with no PV-PQ bus-type switching, since generator re-

active capabilities are incorporated in the functional operating

constraints and no local control adjustments, such as tap and

switchable shunts [31], since the settings of these controls are

determined by the GA. Therefore, only a few load flow itera-

tions are required for convergence. The FDLF and ma-trices areformed and factorized only once in the beginningthe

effect of the changes of shunt admittances on the matrix is

neglected. In case that, due to the random (yet within limits)

initial selection of the control variables, the load flow does not

converge within a predefined number of iterations (set to 8),

large penalty terms, proportional to the maximum active/reac-

tive power mismatch, are added to the FF.

C. Advanced and Problem-Specific Genetic Operators

One of the most important issues in the genetic evolution

is the effective rearrangement of the genotype information. In

the SGA crossover is the main genetic operator responsiblefor the exploitation of information while mutation brings new

nonexistent bit structures. It is widely recognized that the

SGA scheme is capable of locating the neighborhood of the

optimal or near-optimal solutions, but, in general, requires

a large number of generations to converge. This problem

becomes more intense for large-scale optimization problems

with difficult search spaces and lengthy chromosomes, where

the possibility for the SGA to get trapped in local optima

increases and the convergence speed of the SGA decreases.

At this point, a suitable combination of the basic, advanced,

and problem-specific genetic operators must be introduced

in order to enhance the performance of the GA. Advanced

Fig. 3. Gene swap operator.

and problem-specific genetic operators usually combine localsearch techniques and expertise derived from the nature of the

problem.

A set of advanced and problem-specific genetic operators has

been added to the SGA in order to increase its convergence

speed and improve the quality of solutions. Our interest was

focused on constructing simple yet powerful enhanced genetic

operators that effectively explore the problem search space. The

advanced features included in our GA implementation are as

follows.

1) Fitness Scaling: In order to avoid early domination of ex-

traordinary strings and to encourage a healthy competi-

tion among equals, a scaling of the fitness of the pop-

ulation is necessary [27]. In our approach, the fitness isscaled by a linear transformation.

2) Elitism: Elitism ensures that the best solution found thus

far is never lost when moving from one generation to an-

other. The best solution of each generation replaces a ran-

domly selected chromosome in the new generation [32].

3) Hill Climbing: In order to increase the GA search speed at

smooth areas of the search space a hill-climbing operator

is introduced, which perturbs a randomly selected control

variable. The modified chromosome is accepted if thereis an increase in FF value; otherwise, the old chromosome

remains unchanged. This operator is applied only to the

best chromosome (elite) of every generation [26], [29].

In addition to the above advanced features, which are called

advanced despite their wide use in most recent GA implemen-

tations to distinguish between the SGA and our EGA, operators

specific to the OPF problem have been added.

All problem-specific operators introduce random modifica-

tion to all chromosomes of a new generation. If the modified

chromosome proves to have better fitness, it replaces the orig-

inal one in the new population. Otherwise, the original chro-

mosome is retained in the new population. All problem-specific

operators are applied with a probability of 0.2. The following

problem-specific operators have been used.

1) Gene Swap Operator (GSO): This operator randomly se-

lects two genes in a chromosome and swaps their values,as shown in Fig. 3. This operator swaps the active power

output of two units, the voltage magnitude of two genera-

tion buses, etc. Swapping among different types of control

variables is not allowed.

2) Gene Cross-Swap Operator (GCSO): The GCSO is a

variant of the GSO. It randomly selects two different

chromosomes from the population and two genes, one

from every selected chromosome, and swaps their values,

as shown in Fig. 4. While crossover exchanges informa-

tion between high-fit chromosomes, the GCSO searches

for alternative alleles, exploiting information stored even

in low-fit strings.
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Fig. 4. Gene cross-swap operator.

Fig. 5. Gene copy operator.

Fig. 6. Gene inverse operator.

Fig. 7. Gene max-min operator.

3) Gene Copy Operator (GCO): This operator randomly se-

lects one gene in a chromosome and with equal proba-

bility copies its value to the predecessor or the successor

gene of the same control type, as shown in Fig. 5. This op-

erator has been introduced in order to force consecutive

controls (e.g., identical units on the same bus) to operate

at the same output level.

4) Gene Inverse Operator (GIO): This operator acts like a

sophisticated mutation operator. It randomly selects one

gene in a chromosome and inverses its bit-values from

one to zero and vice versa, as shown in Fig. 6. The GIO

searches for bit-structures of improved performance, ex-

ploits new areas of the search space far away from the cur-

rent solution, and retains the diversity of the population.

5) Gene Max-Min Operator (GMMO): The GMMO tries to

identify binding control variable upper/lower limit con-

straints. It selects a random gene in a chromosome and,

with the same probability (0.5), fills its area with 1 s or

0 s, as shown in Fig. 7.

D. Enhanced Genetic Algorithm (EGA)

In the EGA, shown in Fig. 8, after the application of the basic

genetic operators (parent selection, crossover, and mutation) the

Fig. 8. Enhanced genetic algorithm (EGA).

advanced and problem-specific operators are applied to producethe new generation.

All chromosomes in the initial population are created at

random (every bit in the chromosome has equal probability

of being switched ON or OFF). Due to the decoding process

selected [(7) and (8)], the corresponding control variables of the

initial population satisfy their upperlower bound or discrete

value constraints (4). However, the initial population candidate

solutions may not satisfy the functional operating constraints

(3) or even the load flow constraints (2) since the random,

within limits, selection of the control variables may lead to load

flow divergence (as already discussed in Section I V-B).

Population statistics computed for the new generation include

maximum, minimum, and average fitness values and the 90%percentile.

Population statistics are then used to adaptively change the

crossover and mutation probabilities [33]. If premature conver-

gence is detected the mutation probability is increased and the

crossover probability is decreased. The contrary happens in the

case of high population diversity.

V. TEST RESULTS

In this section, the proposed EGA solution of the OPF is eval-

uated using two test systems: 1) the IEEE 30-bus system [6] and

2) the 3-area IEEE RTS96 [34]. The test examples have been

run on a 1.4-GHz Pentium-IV PC. Twenty runs have been per-

formed for each case examined. The results which follow are

the best solution over these 20 runs.

A. IEEE 30-Bus System

The first test system is the IEEE 30-bus, 41-branch system

[6]. It has a total of 24 control variables as follows: five unit ac-

tive power outputs, six generator-bus voltage magnitudes, four

transformer-tap settings, and nine bus shunt admittances.

The gene length for unit power outputs is 12 bits and for gen-

erator voltage magnitudes is 8 bits. They are both treated as con-

tinuous controls. The transformer-tap settings can take 17 dis-

crete values (each one is encoded using 5 bits): the lower and
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Fig. 9. FF comparison for IEEE 30-bus system.

upper limits are 0.9 p.u. and 1.1 p.u., respectively, and the step

size is 0.0125 p.u. The bus shunt admittances can take six dis-

crete values (each one is encoded using 3 bits): the lower and

upper limits are 0.0 p.u. and 0.05 p.u., respectively, and the step

is 0.01 p.u. (on system MVA basis). The GA population size is

taken equal to 80, the maximum number of generations is 200,and crossover and mutation are applied with initial probability

0.9 and 0.001, respectively.

Two sets of 20 test runs were performed; the first (SGA) with

only the basic GA operatorsand the second (EGA) with all oper-

ators, including advanced and problem-specific operators. The

FF evolution of the best of these runs is shown in Fig. 9. The

best and worst solutions of the second set of 20 runs (EGA)

are shown in Table I. The operating costs of the best and worst

solutions are 802.06 $/h and 802.14 $/h, respectively, (0.01%

difference). The differences between the values of the control

variables in the best and worst solutions are not significant. The

operating cost of all EGA-OPF solutions is slightly less than the

802.4 $/h figure reported in [6]. As shown in Table I, there is a

slight difference in unit marginal costs (UMCs), attributed to

network losses. Note that, in this case, the UMCs coincide with

the nodal prices, since no unit limits are reached.

Fig. 9 demonstrates the improvement achieved with the inclu-

sion of the advanced and problem-specific operators. The SGA

run took 18 s, while the EGA took 76 s to evaluate 200 gen-

erations. However, the EGA provides a far better solution than

SGA even in the first 25 generations, or 10 s.

B. IEEE 3-Area RTS96

The 3-area IEEE RTS-96 [34] is a 73-bus, 120-branch system.

It consists of three areas connected through five tie lines. Thearea-A unit cost data are derived from the heat rate data pro-

vided in [34] and the fuel cost data listed in Table II. The value

of water is zero, assuming excessive inflows. Area-B and area-C

fuel costs are selected three times the area-A fuel costs, to im-

pose exports from area A to areas B and C. A contingency case

with tie lines 107203 and 123217 out of service, under 90%

peak load conditions, is studied. To impose congestion, the rat-

ings of tie lines 113215 and 121325 are reduced by 50% (to

250 MVA).

This system has a total of 150 control variables as follows:

98 unit active power outputs, 33 generator-bus voltage magni-

tudes, 16 transformer tap-settings, and 3 bus shunt admittances.

TABLE IIEEE 30-BUS SYSTEM RESULTS

TABLE IIFUEL COSTS FOR IEEE 3-AREA RTS-96

The lower and upper limits of voltage magnitude of all buses are

0.95 p.u. and 1.05 p.u., respectively, (except for PV buses where

p.u.). Transformer taps take discrete values within

0.9 p.u. and 1.1 p.u. with a step size of 0.0125 p.u (17 discretevalues). Similarly, bus shunt admittances take discrete values

between 150 MVAR (inductor, at rated voltage) and 0 MVAR

with a 50 MVAR step (four discrete values). The GA popula-

tion size is taken equal to 180, the maximum number of genera-

tions is 600, and crossover and mutation are applied with initial

probability 0.9 and 0.001, respectively. It was necessary to in-

crease both the population size and the maximum number of

generations to solve the larger problem. It was also necessary to

increase the probability of application of problem-specific op-

erators from 0.2 to 0.5.

First, the unconstrained schedule is obtained by ignoring

branch flow limits. Branch flow limits are ignored by selecting

the corresponding penalty weight to zero in (9). The uncon-strained schedule results in an 81.8 MVA overloading of tie line

121325. The corresponding operating cost is 255 281.5 $/h.

Next, the constrained schedule is calculated by activating the

branch flow constraints. Tie line 121325 flow is now reduced

to 249.97 MVA (almost to the 250 MVA line rating). The

operating cost is increased to 256 189.4 $/h due to congestion.

The FF evolution of both the SGA and the EGA, shown in

Fig. 10, demonstrates the improvement achieved with the inclu-

sion of the advanced and the problem-specific operators. The

SGA run took 266 s, while the EGA took 1643 s to evaluate 600

generations. However, as shown in Fig. 10, a far better solution

is provided by EGA during the same execution time as SGA.
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Fig. 10. FF comparison for IEEE 3-area RTS96.

VI. COMPUTATIONAL REQUIREMENTS OF GENETIC

ALGORITHM OPTIMAL POWER FLOW

In an SGA, if a population of size PS is allowed to evolve

for a total number of NG generations, the product PG NG

determines the required number of FF evaluations (NFE) and

hence the GA computational requirements. When problem-spe-

cific operators are used, the required number of fitness evalua-

tions increases accordingly. In GA-OPF, the FF evaluation is

synonymous with power flow solution in terms of computa-

tional requirements, since the latter is the computationally dom-

inant task in the FF evaluation procedure (see Section IV-B).

It is widely recognized among GA practitioners that the re-

quired NFE for a particular GA implementation depends on

problem difficulty, which, in turn, depends on two factors: 1)

the chromosome length and 2) the shape and characteristics of

the fitness landscape. Problems with smooth fitness landscapes

are easy to solve with GA. If the global optimum is located at

the bottom of a steep gorge of the fitness landscape, the GA mayrequire a large number of fitness evaluations to locate it. Thus,

two optimization problems with the same chromosome length

may require vastly different NFE to solve owing to the differ-

ence in their fitness landscapes.

In GA-OPF, the chromosome length is determined by the

number of control variables and the resolution required for each

control type. The number of buses affects the fitness evalua-

tion ( power flow solution) time. The fitness landscape of the

GA-OPF is very hard to visualize, except for trivial problems

employing at most two-decision variables.

For the assessment of the GA-OPF computational require-

ments, an experiment is designed as follows. Four test systems

of increasing size are created, based on the IEEE RTS96 [34](1-, 3-, 5-, and 10-area configurations). The GA population size

is 200, the probability of application of problem-specific opera-

tors is 0.5, and the maximum number of generations is 600 in all

four cases. Table III summarizes the results of 20 test runs in all

test systems. The last four columns report the average (over the

20 runs) computational requirements of the GA. The number

of generations (NG) to arrive at a good quality OPF solution

is reported in the 7th row. A good quality OPF solution is one

with fitness value within 0.1% of the fitness obtained after al-

lowing the GA to evolve for 600 generations (well within the

flat portion of Fig. 10). The execution time to arrive at a good

quality solution is reported in the9th row. Theresults of Table III

TABLE IIICOMPUTATIONAL REQUIREMENTS

show that thedifference of the best and worst solutions increases

slightly and the execution time increases considerably as the

system size increases.

VII. CONCLUSIONS

A GA solution to the OPF problem has been presented and

applied to small and medium size power systems. The main ad-vantage of the GA solution to the OPF problem is its modeling

flexibility: nonconvex unit cost functions, prohibited unit oper-

ating zones, discrete control variables, and complex, nonlinear

constraints can be easily modeled. Another advantage is that it

can be easily coded to work on parallel computers.

The main disadvantage of GAs is that they are stochastic

algorithms and the solution they provide to the OPF problem

is not guaranteed to be the optimum. Another disadvantage is

that the execution time and the quality of the provided solu-

tion deteriorate with the increase of the chromosome length, i.e.,

the OPF problem size. The applicability of the GA solution to

large-scale OPF problems of systems with several thousands of

nodes, utilizing the strength of parallel computers, has yet to be

demonstrated.

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Anastasios G. Bakirtzis (S77M79SM95) received the Dipl. Mech. &Electr. Eng. degree from the National Technical University of Athens, Athens,Greece, in 1979, and the M.S.E.E. and Ph.D. degrees from Georgia Institute ofTechnology, Atlanta, in 1981 and 1984, respectively.

In 1984, he was a consultant to Southern Company, Atlanta, GA. Since 1986,he has been with the Department of Electrical Engineering, Aristotle Univer-sity of Thessaloniki, Thessaloniki, Greece, where he is currently an AssociateProfessor. His research interests are in power system operation and control, re-liability analysis, and alternative energy sources.

Pandel N. Biskas (S01) received the Dipl. Electr. Eng. degree from the Aris-totle Universityof Thessaloniki, Thessaloniki, Greece, in 1999, where he is cur-rently pursuing the Ph.D. degree. His research interests are in power system op-eration and control and transmission pricing.

Christoforos E. Zoumas (S98) received the Dipl. Electr. Eng. degree fromthe Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996, wherehe is currently pursuing the Ph.D. degree. His research interest is in computerapplications in power systems.

Vasilios Petridis (M77) received the B.S. degree in electrical engineering fromthe National Technical University, Athens, Greece, in 1969, and the M.Sc. and

Ph.D. degrees in electronics and systems from Kings College, University ofLondon, London, U.K., in 1970 and 1974, respectively.Hehas been Consultant ofthe Naval Research Centre, Greece,andDirector of

the Department of Electronics and Computer Engineering and Vice-Chairmanof the Faculty of Electrical and Computer Engineering at Aristotle University,Thessaloniki, Greece. He is currently Professorin the Departmentof Electronicsand Computer Engineering in the Aristotle University of Thessaloniki, Thessa-loniki, Greece. He is coauthor of the monograph Predictive Modular Neural

Networks: Application to Time Series (Norwell, MA: Kluwer, 1998). He is alsoauthor of four books on control and measurement systems and approximately110 research papers. His research interests include control systems, machinelearning, intelligent and autonomous systems, artificial neural networks, evo-lutionary algorithms, fuzzy systems, modeling and identification, robotics, andindustrial automation.

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