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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 2, MAY 2006
An Initialization Procedure in Solving Optimal
Power Flow by Genetic Algorithm
Mirko Todorovski and Dragoslav Rajičić, Senior Member, IEEE
Abstract—The recently published idea of treating voltage angles
at generator-buses as control variables enables to obtain voltages
at load-buses with less computation. However, application of this
approach in solving the optimal power flow problem by genetic algorithms may be ineffective if starting values of voltage angles are
selected quite randomly. To overcome these difficulties, a new procedure for selection of an initial set of complex voltages at generator-buses is proposed in this paper. With this procedure, one can
start the optimization process (i.e., genetic algorithm) with a set of
control variables, causing few or no violations of constraints. The
application of voltage angles at generator-buses as control variables and the proposed initialization procedure is illustrated on
the IEEE test systems. The obtained results are analyzed and compared with the results from the literature. They are competitive,
with computational time drastically reduced.
Index Terms—Bus admittance matrix, generator-bus modeling,
genetic algorithms (GAs), load flow analysis, optimal power flow
(OPF).
I. INTRODUCTION
T
HE minimization of total fuel costs, referred to as economic dispatch [1] or optimal power flow problem (OPF)
[2], is one of the ever-actual power system problems. It has been
a subject of intense power system research for more than four
decades, resulting in many relevant publications. In particular,
under the deregulated environment in the electricity industry in
the past few years, the interest in OPF has become even more
pronounced.
Many optimization techniques have been adapted and used
to solve OPF. A review of selected OPF literature until 1993
can be found in [3] and [4], having the applied techniques
classified as nonlinear programming, quadratic programming,
Newton-based solution, linear programming, hybrid versions
of linear programming and integer programming, and interior
point method. The continuous research in this field has led
to many new contributions beyond 1993. Some of the previously used approaches have been modified and improved (e.g.,
[5]–[9]), and some other techniques have been used, such as
simulated annealing [10], genetic algorithms (GAs) [12]–[16],
neural networks [17]–[20], dual-type method [21], [22], mean
field theory [23], evolutionary programming [24]–[27], tabu
search algorithm [28], particle swarm optimization [29], and
Manuscript received February 15, 2005; revised October 6, 2005. Paper no.
TPWRS-00088-2005.
M. Todorovski is with the Research Center for Energy, Informatics and Materials, Macedonian Academy of Sciences and Arts, Skopje, Republic of Macedonia (e-mail: [email protected]; [email protected]).
D. Rajičić is with the University “Sv. Kiril i Metodij,” Skopje, Republic of
Macedonia (e-mail:[email protected]).
Digital Object Identifier 10.1109/TPWRS.2006.873120
ordinal optimization theory [30]. Many of these techniques
overcome the difficulties in the modeling of complicated cost
functions, discrete control variables, and prohibited unit-operating zones. Some deficiencies in OPF are elaborated on in
[31], and many challenges are discussed in [32].
A specific procedure for initialization and treatment of the
voltage angles at generator-buses as control variables are the
main components of the approach for solving OPF proposed in
this paper.
In order to test the proposed approach, it was applied to
solve OPF by genetic algorithms (GA-OPF). However, this
approach could also be applied in other methods based on similar principles as GAs and evolutionary programming. These
methods have the ability to handle any type of the objective
function, variables, and constraints. Computation procedures
of these methods offer not one “ideal” solution but rather a set
of applicable near-optimal solutions, and they are suitable for
parallel computation.
II. PROBLEM FORMULATION CONSIDERING POWER FLOW
REQUIREMENTS WITHIN GA
The OPF is a constrained optimization problem requiring
minimization of an objective function, which is the total power
generation cost
(1)
subject to
(2)
(3)
where is the real power output of unit , is the cost function
is the total number of units.
of unit , and
The equality constraints (2) are the power flow equations,
while the inequality constraints (3) are due to various limitations. The limitations include lower and upper limits on generator real and reactive powers (respecting possible prohibited
zones as well), limits on voltage magnitudes, line and transformer maximum currents, and sets of possible transformer taps
position and shunt admittances.
and control
variables, which should
The sets of state
satisfy all constraints, are somewhat different than usual in this
paper. It is well known that four quantities are assigned to each
of the buses in a power network. These quantities are injected
real power, injected reactive power, voltage magnitude, and
voltage angle. In the OPF, the only scheduled bus quantities are
0885-8950/$20.00 © 2006 IEEE
TODOROVSKI AND RAJIČIĆ: INITIALIZATION PROCEDURE IN SOLVING OPF BY GA
the injected real and reactive power at load-buses, and none of
the four bus quantities are scheduled at generator-buses.
The generator-bus treatment influences, to a great extent, both
speed and robustness of any power flow method, and since the
power flow calculations are dominating within the GA-OPF approach, appropriate attention should be paid to the subject. As
suggested in [34], complex voltages at generator-buses may be
taken as control variables. In such a way, the concept of one
slack bus is abandoned. This is in accordance with the nature
of the OPF, since one cannot know in advance which is the
best slack bus selection. Therefore, in the proposed approach,
the generating unit’s real and reactive power output are state
variables, which should satisfy(2) and (3). Introducing many
buses), the number
buses with known complex voltages (
of unknown complex voltages is smaller, and the conditions in
the network enable to solve the power flow problem with less
computations. As a result, the whole GA-OPF procedure is less
time-consuming.
In this paper, the set of state variables includes generator
outputs (real and reactive), load-bus voltages, line currents, and
transformer currents. The set of control variables consists of
complex voltages at generator-buses, reactive powers of synchronous condensers, transformer tap settings, and shunt devices settings.
III. GAs
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1) Selection of Initial Generator Power Outputs: Assume
that the operating costs of unit can be represented by
(4)
are cost coefficients. In cases where opwhere , , and
erating costs are not represented by a quadratic function, we
could approximate them by such function. The approximation
will only be used for getting the initial power outputs of the units
and will not be used in the objective function. In addition, we
assume that all generators are online and connected in one point.
is equal to the sum of loads plus
The total power demand
. For zero-cost units (i.e.,
power losses in the network
hydro power plants), we take corresponding scheduled power
outputs. Then, by the subroutine LCONG from the software
package Fortran PowerStation 4.0 [38], which minimizes a general objective function subject to linear equality/inequality con,
(i.e.,
straints, we obtain generator power output ,
“economic dispatch solution”), as a solution of (1) subject to
(5)
(6)
where TPP and HPP are sets of the thermal and hydro power
plants, respectively. In addition, we calculate weightings
A. Problem Encoding
Each control variable is called a gene, while all control variables integrated into one vector is called a chromosome.
In this paper, we use real-coded GA where each chromosome
consists of four regions, one for each subset of control variables.
Those subsets are generator-bus voltage magnitudes and angles,
synchronous condensers reactive powers, transformer tap settings, and shunt admittances.
The GA always deals with a set of chromosomes called a
population. Transforming chromosomes from a population, we
obtain a new population, i.e., next generation. To do this, we use
three genetic operators: selection, crossover, and mutation.
B. Initialization
Usually, at the beginning of the GA optimization process,
each variable gets a random value from its predefined domain.
However, this very simple initialization procedure was found
insufficient for the GA-OPF approach where generator-bus
voltage angles are taken as control variables. In fact, in more
complex networks, the GA-OPF procedure with randomly
initiated voltage magnitudes and angles may not produce a
feasible solution, even in hundreds of generations. Therefore, it
is reasonable to make a special initialization procedure in which
the knowledge of the power systems will be incorporated.
Since the generator power outputs have well-defined lower
and upper limits, while the voltage angles do not, in the initialization procedure (and nowhere else), we select the real power
outputs and voltage magnitudes at generator buses. Afterwards,
the corresponding voltage angles are calculated.
Following this idea, we developed a practical initialization
procedure, which will be explained in this section.
(7)
(8)
where
.
chromosomes of the population are divided into
All
three subsets. The first subset contains only one chromosome.
The generator power outputs obtained by the LCONG optimization procedure are assigned to this chromosome. The second
chromosomes as well as the third.
subset contains
For each chromosome of the second subset, see the following.
1) Randomly select unit using roulette selection method
and weightings obtained by (7).1
2) If
is equal to lower or upper limit, then repeat the
selection; otherwise, set power output of the selected unit
to the lower limit.
3) If spinning reserve of the remaining units is lower than a
certain predefined value (e.g., 10%), then repeat the selection.
4) Calculate power outputs of the other units by the LCONG
optimization procedure.
For each chromosome of the third subset, see the following.
1) Randomly select unit using roulette method and weightings obtained by (8).2
1Units
2Units
having higher production costs are more likely to be selected.
having lower production costs are more likely to be selected.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 2, MAY 2006
2) If
is equal to lower or upper limit, then repeat the
selection; otherwise, set power output of the selected unit
to the upper limit.
3) Calculate power outputs of the other units by the LCONG
optimization procedure.
2) Selection of Initial Voltage Magnitudes: We split the allowable generator-bus voltage magnitude interval into number
of levels equal to the number of chromosomes in the population.
Therefore, each voltage level corresponds to one of the chromosomes. Then, within each of the chromosomes, we assign corresponding voltage level to all generator-buses. This procedure
is referred to as “voltage-grating.”
3) Selection of Initial Taps and Shunts Settings: The initial
values of the rest of the control variables are selected at random
from its predefined domain.
4) Proposed Initialization Procedure: The initial values of
voltage angles and magnitudes at generator-buses can be obtained in the following steps.
1) Select the initial generator real power outputs.
2) For each of the chromosomes, assign the corresponding voltage level to generator-buses using the
“voltage-grating” procedure (introduced in this subsection).
3) For each chromosome, do the following.
3.1) Calculate voltage angles at generator-buses and
complex voltages at load-buses by the fast decoupled or Newton’s method. In this calculation, we
use generator real power outputs from step 1), generator-bus voltage magnitudes from step 2), and
scheduled real and reactive loads.
3.2) Check for violated generator reactive power outputs. If violation occurs, reduce/increase the corresponding bus voltage magnitude and go to step 3.1).
Otherwise, accept the actual voltage magnitudes and
angles at generator-buses as initial values for the
chromosome.
At this point, it might seem that step 3.1) is a deviation from
the main idea of this paper, which is the use of generator-bus
voltage angles as control variables. Nevertheless, it should be
emphasized that this is only an intermediate step toward population initialization.
C. Chromosome Fitness
Fitness is a quantity related to the chromosome. It serves to
enable comparison between chromosomes. At the stage when
we calculate the fitness, we have already solved the power flow
equations, meaning that the constraints (2) are satisfied. However, all constraints (3) have to be checked for violations. In this
paper, the penalty method is used, which degrades the fitness in
cases with violated constraint. The fitness for chromosome is
defined by
(9)
where
is the objective function related to chromosome
[as in (1)],
is the set of violated constraints associated to
is the penalty term corresponding to conchromosome , and
straint .
For example, if variable (having upper limit and lower
) is of type , for which the penalty coefficient is , then in
a case of constraint violation, the corresponding penalty term
included in (9) will be
if
if
(10)
For the set of violated constraints, four different penalty coefficients are used, related to the following state variables: generator real and reactive powers, voltage magnitudes, and branch
MVA flows. As suggested in [2], it is quite effective to start
with low values of penalty coefficients and to increase them
during the optimization process. The penalty increase should be
controlled; otherwise, it may be counterproductive and perform
worse than the case with constant penalty coefficients. Consequently, the following control scheme is applied. After a certain number of generations (usually ten), we check whether the
best chromosome has changed since the last check and whether
there are violated constraints related to it. If in both cases the answers are positive, then the corresponding penalty coefficients
are multiplied by a certain factor (usually two). In addition, it is
advisable not to let the penalty coefficients increase too much.
Note that in this approach, within each generation of the GA
solution process, for each of the chromosomes, we evaluate
complex voltages at generator-buses by GA and then calculate
complex voltages at load-buses. With these voltages, we can directly calculate injected power at each of the generator-buses,
regardless of the number of generators at the bus. However, two
or more generators can exist at some buses, in which case, the
share of each generator to the total injected power should be
determined. This can even be done prior to the GA-OPF optimization by solving a small problem of a few parallel-connected generators at one point. In such a way, we can construct
a lookup table with parallel generator power outputs offering
minimal production costs.
D. Selection
Improvement of the average fitness of the population is
achieved through selection of individuals as parents from the
completed population. The selection is performed in such a
way that chromosomes having higher fitness are more likely to
be selected as parents.
Bearing in mind that some of the individuals may have significantly higher fitness than the others, the next generation may be
constituted of large number of identical individuals. This poses
a limitation on the population diversity, meaning that the search
space will be reduced. One can omit such a situation by using
relative fitness [11]. Let
and
be the minimal and the
maximal fitness in the population, respectively. Then the relative fitness for chromosome is defined as
(11)
TODOROVSKI AND RAJIČIĆ: INITIALIZATION PROCEDURE IN SOLVING OPF BY GA
483
There are several selection techniques. The roulette selection
method is used here. In this method, we calculate the relative
weight of each chromosome’s fitness as
(12)
Therefore, the likelihood of selecting a chromosome as a
parent is a function of its fitness relative to the total.
To further improve the evolving process, the GA can carry
over the best individuals from the completed population to the
new population set (principle of elitism).
E. Crossover
After the selection, the GA picks a pair of selected chromosomes in order to create two new chromosomes. The GA applies a random generator to cut the strings at any position (the
crossover point) and exchanges the substrings between the two
chromosomes. After the crossover is performed, the new chromosomes are added to the new population set.
F. Mutation
The mutation is specifically applied to increase population
diversity. Mutation involves randomly selecting genes within
the chromosomes and assigning them random values within the
corresponding predefined interval. In order not to destroy good
genetic code, nonuniform mutation has to be applied. In such
a manner, in later stages of GA optimization process, the interval for random selection of genes’ values is narrowed [11]. In
the course of mutation, gene will take a new value depending
on generation number , maximum number of generations ,
nonuniform mutation parameter , and two random numbers
and
as follows:
if
(13)
if
where and are gene’s minimal and maximal values (from
the permissible domain).
The probability of mutation is normally kept very low, as high
mutation rates could degrade the evolving process into a random
search process.
G. GA Parameters
GA requires definition of a number of parameters, which can
affect the efficiency of the search process in several ways.
The population size
should be large enough to create
sufficient diversity covering the possible solution space. Genin advance.
erally, one cannot know the optimal value of
Clearly, a more complex problem domain requires a larger
due to larger possible combination of variables.
Another user-defined criterion is the point at which the optimization process terminates. In this paper, we use GA with fixed
Fig. 1.
Flowchart of the proposed optimization procedure.
number of generations, for which we assume that the search
process has covered sufficient search space.
Other parameters, such as crossover probability, mutation
rate, selection, and crossover mechanisms, seem to affect the
GA process less significantly when evaluated over a larger
number of generations.
The flowchart of the proposed optimization procedure is
shown in Fig. 1.
IV. RESULTS AND DISCUSSION
The proposed approach was applied on the following test systems: IEEE 30 [25], [27], [39], IEEE 118 [26], [39], 1-area IEEE
RTS96 [15], [39], and 3-area IEEE RTS96 [15], [39]. In some
cases, the results obtained by the proposed o influence on the
final result. approach are compared with the results from the
literature. In all those cases, the crossover and mutation probability, as well as the population size and number of generations,
were taken from the corresponding papers. The nonuniform mutation parameter in (13) is set to 5. The results were used to
investigate evolution of the objective function, final objective
function values, computation times, and method robustness.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 2, MAY 2006
TABLE I
SOLUTION FOR THE IEEE 30-BUS TEST SYSTEM
Fig. 2. Objective function evolution—IEEE 30.
TABLE II
FINAL RESULTS OF THE TESTS
Fig. 3. Objective function evolution—1-area IEEE RTS96.
In some of the figures, we can see intervals in which the objective function increases. They appear when there are overloaded
lines as a consequence of the GA activity to eliminate the violations.
B. Final Results
Fig. 4. Objective function evolution—IEEE 118.
Fig. 5. Objective function evolution—3-area IEEE RTS96.
A. Evolution of the Objective Function
Figs. 2–5 illustrate the evolution of the best objective function values through generations. In these figures, there are two
lines representing two different initialization procedures: solid
line for the initialization proposed in this paper (see Section III)
and dotted line for the standard initialization procedure (random
selection of real powers and voltage magnitudes). These figures
show that appropriate reduction of the number of generations
has almost n
Table I presents one solution for the IEEE 30-bus test system,
containing the genes’ values (voltage magnitudes and angles)
along with the generators’ real and reactive power outputs.
From the OPF solution, the power system operator can get the
injected real power and voltage magnitude for every generator,
as well as transformer taps, and shunt admittances settings. In
other words, he can set the system in optimal state by adjusting
the real power output through the governor loop and voltage
magnitude through the exciter loop. Also, Table I contains the
production costs for all generators.
In Table II, we present the final values of the objective function obtained by the proposed approach (20 runs), along with
the results reported in the corresponding papers. The table contains the best solutions, as well as the average values of all 20
runs. A good concordance in the results is evident.
C. Computation Times
In Table III, the time consumption (measured on AMD
Athlon 1,833 MHz) for the whole GA-OPF run for two different cases is presented. In both cases, we used the same
GA-OPF approach explained in this paper, but we calculated
voltages at load-buses by different methods. In case 1, we
applied the power flow method from [34] (see the Appendix),
while in case 2, we applied the fast decoupled power flow
method from [33]. In both cases, we used the real and reactive
power mismatches as a termination criterion for the voltage calculation procedure. In addition, the same seed for the random
TODOROVSKI AND RAJIČIĆ: INITIALIZATION PROCEDURE IN SOLVING OPF BY GA
TABLE III
TIME CONSUMPTIONS USING DIFFERENT POWER FLOW METHODS
TABLE IV
RESULTS WHEN CHROMOSOMES INCLUDE GENERATOR
REAL POWERS INSTEAD OF VOLTAGE ANGLES
485
V. CONCLUSION
In this paper, a novel approach in solving the OPF problem
by GA has been proposed. It is based on the application of
the new initialization procedure and recently published idea of
using voltage angles at generator-buses as control variables.
The application of the proposed approach to several IEEE test
systems shows that the proposed initialization procedure improves the performance of the whole GA-OPF procedure. Numerous tests illustrate that the proposed approach gives results
competitive to the results obtained by corresponding methods
from the literature. In addition, the computational time is drastically reduced.
APPENDIX
A. Review of the Power Flow Method From [34]
TABLE V
RESULTS FOR THE IEEE 30-BUS TEST SYSTEM
In this approach, it is assumed that complex voltages at all
generator-buses are known. The complex bus admittance mais formed taking into account initial values of the
trix
transformer turn ratios and shunt admittances.
, for load-bus , we can write
Using the elements of
(14)
(15)
(16)
number generator was applied, so we are certain that the initial
populations are identical, as well as the evolution path. In such
a way, we produced exactly the same final results.
In Table IV, we present the results (time consumption as an
average of all 20 runs and the best generation costs) for the
cases where the generator real powers and voltage magnitudes
are selected as control variables (i.e., as genes). In case 1, the
initial generator real power outputs and voltage magnitudes are
selected randomly, while in case 2, the procedure proposed in
this paper was applied. In both cases, the fast decoupled power
flow method was employed. The final results are pretty similar
to the results of Table II and, in the case when the proposed
initialization was used, are slightly better. As it can be seen, the
difference between the running times from Table III, case 1, and
Table IV is obvious.
D. Robustness
In order to investigate the robustness of the proposed approach, it is applied to the IEEE 30-bus test system using the
following three types of generator cost curves (as in [25]):
quadratic, piecewise quadratic, and quadratic with a sine
component superimposed upon it. In the last type, the sine
component is used to represent the valve-point loading effect
[12].
The obtained results are presented in Table V, where a good
match with the results from [25] is obvious.
where
is the complex load at bus ,
is the difference between the actual and initial values of complex shunt
is the complex voltage at bus ,
admittance at bus ,
is the sum of currents modeling tap changes [as in (23)] at all
is the element of
in
transformers connected to bus ,
and
are sets of indexes of
row and column , and
load-buses and generator-buses directly connected to bus , respectively. Asterisk as a superscript denotes complex conjugate.
Writing (14) for every load-bus of the system, a set of simultaneous equations is obtained. It can be expressed in matrix form
as
(17)
and
are calculated from
where elements of vectors
(15) and (16), respectively. At the beginning of the voltage calcan be obtained from
by exculation procedure,
cluding rows and columns related to generator-buses.
Load-bus voltages can be calculated from (17) by an iterative
depend on the load-bus
procedure, since the elements of
voltages that are not known at the beginning. On the other hand,
do not change from iteration to iteration,
the elements of
and we calculate them only once. In addition, application of the
transformer model explained in Subsection B of this Appendix
constant in spite of changes in transformer
allows keeping
should be formed and factorized
tap settings. As a result,
only once.
486
For
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 2, MAY 2006
and
in two successive iterations, we can define
(18)
Fig. 6.
and
Transformer equivalent circuit.
(19)
Then, from (17)–(19), we obtain
(20)
Fig. 7. Representation of the transformer from Fig. 6.
is sparse, the use of sparse matrix factorization
Since
is recommended when solving (17) and (20). In addition, it is
beneficial to apply the method proposed by Tinney and Walker
for structurally symmetric matrices [35], as well as Markowitz’s
strategy and Duff’s search technique [36]. In this procedure,
the application of row-column permutation contributes to fill-ins
minimization, while the threshold pivoting strategy improves
the numerical stability.
At the beginning of the GA-OPF procedure, the matrix
is created and factorized. The iterative procedure for calculation
of load-bus voltages consists of the following steps.
per unit (flat start).
1) Set all load-bus voltages at
[using (15) and (23)] and
2) Calculate the elements of
[using (16)].
the elements of
3) Solve (17) and get new load-bus voltages.
[using (15) and (23)] and
4) Calculate new elements of
[using (19)].
the elements of
and update
.
5) Solve (20) for
are not less than
6) If magnitudes of all elements of
the specified voltage tolerance, go to step 4); otherwise,
the iterative process is finished.3
B. Network-Model
It should be noted that the selection of voltage angles at
generator-buses as control variables enables the construction
of a network-model, which can be used in calculation of voltages at load-buses. The network-model does not contain the
generator-buses. It can be obtained from the original network
making the following changes: each branch connecting a
generator-bus with a load-bus should be substituted by a shunt
branch and shunt current generator at the load-bus. The shunt
branch impedance is equal to the impedance of the substituted
branch. The current of the current generator is equal to the
quotient of the voltage at the generator-bus and the impedance
can be obtained as the
of the substituted branch. Then,
bus admittance matrix of the network-model.
It should be noted that in some cases, network-models consist
of several parts. Voltage calculations in each of those parts can
be done separately. In addition, some of the parts can have radial
or weakly meshed topological structure. In those cases, load-bus
voltages can be calculated by using methods effective for radial
and weakly meshed networks.
C. Handling Changes of Transformer Taps
Let transformer have complex turns ratio
and admittance
, and connect buses and . Corresponding equivalent circuit
contains an ideal transformer with complex turns ratio
:1 in
series with admittance (see Fig. 6). The bus admittance matrix of the transformer from Fig. 6 is (e.g., [37])
(21)
are
According to (21), three out of four elements of
keeps changing as the GA searches
not constant, since ratio
for the optimal solution. This could pose serious drawbacks on
the voltage calculation procedure, because if one goes straightwill be required.
forward, many factorizations of
This problem can be bypassed following a different calculation path, in which the influence of the ratio changes can be
modeled by current injections. To do this, for every transformer
, we first calculate the transformer admittance matrix
using the initial value of its turns ratio
and put it into
.
Then, in cases when turns ratio of the transformer changes to
, corresponding admittance matrix
can be repreand the matrix increment
sented as a sum of
(22)
constant (and avoid additional factorIn order to keep
izations), we do not put
into
, but we simulate it in
(20) by injected currents
(23)
3In
order to have the same termination criterion for both cases in Table III,
the procedure does not stop after step 6) but rather continues with step 7):
7) Calculate power mismatches ( P and Q ). If magnitude of any power
mismatches is greater than the specified power tolerance, then continue as in
step 4); otherwise, the iterative process is finished.
1
1
that should be subtracted from
Fig. 7).
and
, respectively (see
TODOROVSKI AND RAJIČIĆ: INITIALIZATION PROCEDURE IN SOLVING OPF BY GA
ACKNOWLEDGMENT
The authors would like to thank Dr. N. Markovska and Ms. S.
Secrest for proofreading the manuscript. Also, the authors specially acknowledge the data provisions for IEEE test systems by
Messrs. M. A. Abido, P. N. Biskas and P. Venkatesh via private
e-mail communication.
REFERENCES
[1] J. Carpentier, Contribution a l’étude du dispatching économique, ser. 8:
Bulletin de la Société Française des Électriciens, Août 1962, vol. III, pp.
431–447.
[2] H. W. Dommel and W. F. Tinney, “Optimal power flow solution,” IEEE
Trans. Power App. Syst., vol. PAS-87, no. 10, pp. 1866–1876, Oct. 1968.
[3] J. A. Momoh, M. E. El-Hawary, and R. Adapa, “A review of selected
optimal power flow literature to 1993, Part 1: Nonlinear and quadratic
programming approaches,” IEEE Trans. Power Syst., vol. 14, no. 1, pp.
96–104, Feb. 1999.
, “A review of selected optimal power flow literature to 1993, Part 2:
[4]
Newton, linear programming and interior point methods,” IEEE Trans.
Power Syst., vol. 14, no. 1, pp. 105–111, Feb. 1999.
[5] Y.-C. Wu, A. S. Debs, and R. E. Marsten, “A direct nonlinear predictorcorrector primal-dual interior point algorithm for optimal power flows,”
IEEE Trans. Power Syst., vol. 9, no. 2, pp. 876–883, May 1994.
[6] G. Torres and V. Quintana, “On a nonlinear multiple-centrality-corrections interior-point method for optimal power flows,” IEEE Trans. Power
Syst., vol. 16, no. 2, pp. 222–228, May 2001.
[7] Y.-C. Wu, “Fuzzy second correction on complementary condition for optimal power flow,” IEEE Trans. Power Syst., vol. 16, no. 3, pp. 360–366,
Aug. 2001.
[8] V. Miranda and J. T. Saraiva, “Fuzzy modeling of power system optimal
load flow,” IEEE Trans. Power Syst., vol. 7, no. 2, pp. 843–849, May
1992.
[9] K. H. Abdul-Rahman and S. M. Shahidehpour, “Static security in power
system operation with fuzzy real load conditions,” IEEE Trans. Power
Systems, vol. 10, no. 1, pp. 77–87, Feb. 1995.
[10] K. P. Wong and C. C. Fung, “Simulated annealing based economic dispatch algorithm,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol.
140, no. 6, pp. 509–515, Nov. 1993.
[11] Z. Michalewicz, Genetic Algorithms+Data Structures=Evolution Programs, 3rd ed. Berlin, Germany: Springer-Verlag, 1999, pp. 111–112.
[12] D. C. Walters and G. B. Sheblé, “Genetic algorithm solution of economic
dispatch with valve point loading,” IEEE Trans. Power Syst., vol. 8, no.
3, pp. 1325–1332, Aug. 1993.
[13] G. B. Sheblé and K. Brittig, “Refined genetic algorithm—economic dispatch example,” IEEE Trans. Power Syst., vol. 10, no. 1, pp. 117–124,
Feb. 1995.
[14] P.-H. Chen and H.-C. Chang, “Large-scale economic dispatch by genetic
algorithm,” IEEE Trans. Power Syst., vol. 10, no. 4, pp. 1919–1926, Nov.
1995.
[15] A. G. Bakirtzis, P. N. Biskas, C. E. Zoumas, and V. Petridis, “Optimal
power flow by enhanced genetic algorithm,” IEEE Trans. Power Syst.,
vol. 17, no. 2, pp. 229–236, May 2002.
[16] T. Yalcinoz, H. Altun, and M. Uzam, “Economic dispatch solution using
a genetic algorithm based on arithmetic crossover,” in Proc. IEEE Porto
Power Tech. Conf., Porto, Portugal, Sep. 2001.
[17] J. H. Park, Y. S. Kim, I. K. Eom, and K. Y. Lee, “Economic load dispatch
for piecewise quadratic cost function using Hopfield neural network,”
IEEE Trans. Power Syst., vol. 8, no. 3, pp. 1030–1038, Aug. 1993.
[18] J. Kumar and G. B. Sheblé, “Clamped state solution of artificial neural
network for real-time economic dispatch,” IEEE Trans. Power Syst., vol.
10, no. 2, pp. 925–931, May 1995.
[19] C.-T. Su and G.-J. Chiou, “A fast-computation Hopfield method to economic dispatch of power systems,” IEEE Trans. Power Syst., vol. 12, no.
4, pp. 1759–1764, Nov. 1997.
[20] T. Yalcinoz and M. J. Short, “Neural networks approach for solving economic dispatch problem with transmission capacity constraints,” IEEE
Trans. Power Syst., vol. 13, no. 2, pp. 307–313, May 1998.
[21] C.-H. Lin and S.-Y. Lin, “A new dual-type method used in solving optimal power flow problems,” IEEE Trans. Power Syst., vol. 12, no. 4, pp.
1667–1675, Nov. 1997.
487
[22] C.-H. Lin, S.-Y. Lin, and S.-S. Lin, “Improvements on the duality based
method used in solving optimal power flow problem,” IEEE Trans.
Power Syst., vol. 17, no. 2, pp. 315–323, May 2002.
[23] L. Chen, H. Suzuki, and K. Katou, “Mean field theory for optimal power
flow,” IEEE Trans. Power Syst., vol. 12, no. 4, pp. 1481–1486, Nov.
1997.
[24] H.-T. Yang, P.-C. Yang, and C.-L. Huang, “Evolutionary programming
based economic dispatch for units with nonsmooth fuel cost function,”
IEEE Trans. Power Syst., vol. 11, no. 1, pp. 112–118, Feb. 1996.
[25] J. Yuryevich and K. P. Wong, “Evolutionary programming based optimal power flow algorithm,” IEEE Trans. Power Syst., vol. 14, no. 4,
pp. 1245–1250, Nov. 1999.
[26] P. Venkatesh, R. Gnanadass, and N. P. Padhy, “Comparison and application of evolutionary techniques to combined economic emission dispatch with line flow constraints,” IEEE Trans. Power Syst., vol. 18, no.
2, pp. 688–697, May 2003.
[27] M. A. Abido, “Environmental/economic power dispatch using multiobjective evolutionary algorithm,” IEEE Trans. Power Syst., vol. 18, no. 4,
pp. 1529–1537, Nov. 2003.
[28] T. Kulworawanichpong and S. Sujitjorn, “Optimal power flow using tabu
search,” IEEE Power Eng. Rev., vol. 22, no. 6, pp. 37–40, Jun. 2002.
[29] Z. L. Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints,” IEEE Trans. Power Syst.,
vol. 18, no. 3, pp. 1187–1195, Aug. 2003.
[30] S.-Y. Lin, Y.-C. Ho, and C.-H. Lin, “An ordinal optimization theorybased algorithm for solving the optimal power flow problem with discrete control variables,” IEEE Trans. Power Syst., vol. 19, no. 1, pp.
276–286, Feb. 2004.
[31] W. F. Tinney, J. M. Bright, K. D. Demeree, and B. A. Hughes, “Some
deficiencies in optimal power flow,” IEEE Trans. Power Syst., vol. 3, no.
2, pp. 676–683, May 1988.
[32] J. A. Momoh, R. J. Koessler, M. S. Bond, B. Stott, D. Sun, A. Papalexopoulos, and P. Ristanović, “Challenges to optimal power flow,” IEEE
Trans. Power Syst., vol. 12, no. 1, pp. 444–455, Feb. 1997.
[33] B. Stott and O. Alsaç, “Fast decoupled load flow,” IEEE Trans. Power
App. Syst., vol. PAS-93, no. 3, pp. 859–869, May/Jun. 1974.
[34] M. Todorovski and D. Rajičić, “A power flow method suitable for
solving OPF problems using genetic algorithms,” in Proc. IEEE Region
8 EUROCON, vol. 2, 2003, pp. 215–219.
[35] W. F. Tinney and J. W. Walker, “Direct solution of sparse network equations by optimally ordered triangular factorization,” Proc. IEEE, vol. 55,
no. 11, pp. 1801–1809, Nov. 1967.
[36] I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse
Matrices. London, U.K.: Oxford Univ. Press, 1986.
[37] G. W. Stagg and A. H. El-Abiad, Computer Methods in Power Systems
Analysis. New York: McGraw-Hill, 1968.
[38] “Fortran PowerStation 4.0,” Microsoft Developer Studio, Microsoft Corporation, 1994–1995.
[39] University of Washington, Department of Electrical Engineering. Power
Systems Test Case Archive. [Online]Available: http://www.ee.washington.edu/research/pstca.
[40] [Online] Available: http://cuaerospace.com/carroll/whats_new_ga.html.
Mirko Todorovski received the B.Sc., M.Sc., and D.Sc. degrees from University “Sv. Kiril i Metodij,” Skopje, Republic of Macedonia, in 1995, 1998, and
2004, respectively.
From 1997–2005, he was with the Research Center for Energy, Informatics,
and Materials of the Macedonian Academy of Arts and Sciences, Skopje, Republic of Macedonia. In 2006, he joined the Faculty of Electrical Engineering
in Skopje, and presently, he is a Teaching Assistant in the Department of Power
Systems. His research interests are related to computer applications in power
system analysis and planning.
Dragoslav Rajičić (SM’97) received the B.Sc. degree from University “Sv.
Kiril i Metodij,” Skopje, Republic of Macedonia, and the M.Sc. and D.Sc. degrees from the University of Belgrade, Serbia and Montenegro, in 1963, 1970,
and 1978, respectively.
He is a retired Professor from the Faculty of Electrical Engineering, University “Sv. Kiril i Metodij.”