IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002
229
Optimal Power Flow by Enhanced Genetic Algorithm
Anastasios G. Bakirtzis, Senior Member, IEEE, Pandel N. Biskas, Student Member, IEEE,
Christoforos E. Zoumas, Student Member, IEEE, and Vasilios Petridis, Member, IEEE
Abstract—This paper presents an enhanced genetic algorithm
(EGA) for the solution of the optimal power flow (OPF) with
both continuous and discrete control variables. The continuous
control variables modeled are unit active power outputs and
generator-bus voltage magnitudes, while the discrete ones are
transformer-tap settings and switchable shunt devices. A number
of functional operating constraints, such as branch flow limits,
load bus voltage magnitude limits, and generator reactive capabilities, are included as penalties in the GA fitness function (FF).
Advanced and problem-specific operators are introduced in order
to enhance the algorithm’s efficiency and accuracy. Numerical
results on two test systems are presented and compared with
results of other approaches.
Index Terms—Genetic 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.
denotes that the entry correA hat above vectors and
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
S
INCE 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 function (such as the operating cost of thermal resources) while satisfying a set of system operating constraints, including constraints
dictated by the electric network. OPF has been widely used in
power 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.
The authors are with the Department of Electrical Engineering, Aristotle University 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 operating 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 contributions 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 problem’s Karush–Kuhn–Tucker (KKT) optimality conditions. 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 to
zero 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 approaches 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-integer
programming, discrete control variables are initially treated
as continuous, and post-processing discretization logic is
subsequently applied [21], [22]. Whereas the effects of discretization 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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Recent attempts to overcome the limitations of the mathematical programming approaches include the application of simulated annealing-type methods [23], [24], and genetic algorithms
(GAs) [25], [26].
In [25], a simple genetic algorithm (SGA) is used for OPF solution. The control variables modeled are generator active power
outputs and voltages, shunt devices, and transformer taps. Branch
flow, reactive generation, and voltage magnitude constraints are
treated as quadratic penalty terms in the GA fitness function (FF).
To keep the GA chromosome size small, only a 4-bit chromosome
area is used for the encoding of each control variable. A sequential GA solution scheme is employed to achieve acceptable control 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 injections 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 devoted
to 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. Variable 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 incorporation of the problem-specific operators, the GA can solve
larger OPF problems. Test results on systems with up to 242
buses 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 optimization problem as follows:
Min
S.t.
(1)
(2)
(3)
(4)
where
(5)
(6)
The equality constraints (2) are the nonlinear power flow
equations. The inequality constraints (3) are the functional operating 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, encoded 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 algorithms. 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 evaluated, genetic evolution takes place by means of three basic genetic operators:
1) parent selection;
2) crossover;
3) mutation.
Parent selection is a simple procedure whereby two chromosomes are selected from the parent population based on their fitness value. Solutions with high fitness values have a high probability of contributing new offspring to the next generation. The
selection rule used in our approach is a simple roulette-wheel
selection [27].
BAKIRTZIS et al.: OPTIMAL POWER FLOW BY ENHANCED GENETIC ALGORITHM
231
Fig. 2. GA chromosome structure.
IV. GENETIC ALGORITHM SOLUTION
OPTIMAL POWER FLOW
TO
A. Encoding
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 convergence speed of the GA and is usually applied with high
probability (0.6–0.9). The chromosomes of the two parents
selected are combined to form new chromosomes that inherit 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 the
information included in the current generation, it does not produce new information.
Mutation is the operator responsible for the injection of new
information. With a small probability, random bits of the offspring 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 iterative procedure, which ends when a maximum number of generations 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]).
The chromosome is formed as shown in Fig. 2. There are
four chromosome regions (one for each set of control vari; 2)
; 3) ; and 4)
. Encoding is
ables), namely, 1)
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)
is the decimal number to which the binary
where
is the gene length
number in a gene is decoded and
(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 objective 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 maximized 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 automatically 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 computed 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 reactive 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 iteraand
mations are required for convergence. The FDLF
trices are formed and factorized only once in the beginning—the
matrix is
effect of the changes of shunt admittances on the
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/reactive 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 responsible
for 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 local
search 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 extraordinary strings and to encourage a healthy competition among equals, a scaling of the fitness of the population is necessary [27]. In our approach, the fitness is
scaled by a linear transformation.
2) Elitism: Elitism ensures that the best solution found thus
far is never lost when moving from one generation to another. The best solution of each generation replaces a randomly 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 there
is 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 implementations to distinguish between the SGA and our EGA, operators
specific to the OPF problem have been added.
All problem-specific operators introduce random modification to all chromosomes of a new generation. If the modified
chromosome proves to have better fitness, it replaces the original one in the new population. Otherwise, the original chromosome 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 selects 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 generation 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 information between high-fit chromosomes, the GCSO searches
for alternative alleles, exploiting information stored even
in low-fit strings.
BAKIRTZIS et al.: OPTIMAL POWER FLOW BY ENHANCED GENETIC ALGORITHM
Fig. 4.
Gene cross-swap operator.
Fig. 5.
Gene copy operator.
233
Fig. 8. Enhanced genetic algorithm (EGA).
Fig. 6. Gene inverse operator.
Fig. 7. Gene max-min operator.
3) Gene Copy Operator (GCO): This operator randomly selects one gene in a chromosome and with equal probability copies its value to the predecessor or the successor
gene of the same control type, as shown in Fig. 5. This operator 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, exploits new areas of the search space far away from the current 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 constraints. 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
advanced and problem-specific operators are applied to produce
the 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 upper–lower 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 convergence 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 evaluated 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 performed 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 active 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 generator voltage magnitudes is 8 bits. They are both treated as continuous controls. The transformer-tap settings can take 17 discrete values (each one is encoded using 5 bits): the lower and
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002
TABLE I
IEEE 30-BUS SYSTEM RESULTS
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 discrete 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 operators and the second (EGA) with all operators, 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 inclusion of the advanced and problem-specific operators. The SGA
run took 18 s, while the EGA took 76 s to evaluate 200 generations. 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. The
area-A unit cost data are derived from the heat rate data provided 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 impose exports from area A to areas B and C. A contingency case
with tie lines 107–203 and 123–217 out of service, under 90%
peak load conditions, is studied. To impose congestion, the ratings of tie lines 113–215 and 121–325 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 magnitudes, 16 transformer tap-settings, and 3 bus shunt admittances.
TABLE II
FUEL 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 discrete
values). 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 population size is taken equal to 180, the maximum number of generations is 600, and crossover and mutation are applied with initial
probability 0.9 and 0.001, respectively. It was necessary to increase 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 operators 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 unconstrained schedule results in an 81.8 MVA overloading of tie line
121–325. The corresponding operating cost is 255 281.5 $/h.
Next, the constrained schedule is calculated by activating the
branch flow constraints. Tie line 121–325 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 inclusion 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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235
TABLE III
COMPUTATIONAL REQUIREMENTS
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-specific operators are used, the required number of fitness evaluations increases accordingly. In GA-OPF, the “FF evaluation” is
synonymous with “power flow solution” in terms of computational requirements, since the latter is the computationally dominant task in the FF evaluation procedure (see Section IV-B).
It is widely recognized among GA practitioners that the required 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 may
require 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 difference 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 evaluation ( 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 requirements, 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 operators 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 allowing 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 the 9th row. The results of Table III
show that the difference 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 advantage of the GA solution to the OPF problem is its modeling
flexibility: nonconvex unit cost functions, prohibited unit operating 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 solution 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 (S’77–M’79–SM’95) 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 of
Technology, 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 University of Thessaloniki, Thessaloniki, Greece, where he is currently an Associate
Professor. His research interests are in power system operation and control, reliability analysis, and alternative energy sources.
Pandel N. Biskas (S’01) received the Dipl. Electr. Eng. degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1999, where he is currently pursuing the Ph.D. degree. His research interests are in power system operation and control and transmission pricing.
Christoforos E. Zoumas (S’98) received the Dipl. Electr. Eng. degree from
the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996, where
he is currently pursuing the Ph.D. degree. His research interest is in computer
applications in power systems.
Vasilios Petridis (M’77) received the B.S. degree in electrical engineering from
the National Technical University, Athens, Greece, in 1969, and the M.Sc. and
Ph.D. degrees in electronics and systems from King’s College, University of
London, London, U.K., in 1970 and 1974, respectively.
He has been Consultant of the Naval Research Centre, Greece, and Director of
the Department of Electronics and Computer Engineering and Vice-Chairman
of the Faculty of Electrical and Computer Engineering at Aristotle University,
Thessaloniki, Greece. He is currently Professor in the Department of Electronics
and Computer Engineering in the Aristotle University of Thessaloniki, Thessaloniki, Greece. He is coauthor of the monograph Predictive Modular Neural
Networks: Application to Time Series (Norwell, MA: Kluwer, 1998). He is also
author of four books on control and measurement systems and approximately
110 research papers. His research interests include control systems, machine
learning, intelligent and autonomous systems, artificial neural networks, evolutionary algorithms, fuzzy systems, modeling and identification, robotics, and
industrial automation.