IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
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A Metaheuristic to Solve the Transmission Expansion Planning
Rubén Romero, Member, IEEE, Marcos J. Rider, Member, IEEE, and Irênio de J. Silva, Student Member, IEEE
Abstract—In this letter, a genetic algorithm (GA) is applied to
solve the static and multistage transmission expansion planning
(TEP) problem. The characteristics of the proposed GA to solve
the TEP problem are presented. Results using some known systems show that the proposed GA solves a smaller number of linear
programming problems in order to find the optimal solutions and
obtains a better solution for the multistage TEP problem.
Index Terms—Combinatorial optimization, genetic algorithm,
metaheuristics, transmission expansion planning.
I. INTRODUCTION
HE transmission expansion planning (TEP) problem of
electrical power systems consists in finding the transmission lines and/or transformers that should be constructed so that
the system can operate in an adequate way and in a specified
planning horizon. Taking into account the planning period, the
planning problem can be considered a one-stage problem (static
planning) or the planning horizon can be separated into several
stages (multistage planning). The mathematical models used for
the static and multistage TEP problem are presented in [1] and
[2], respectively. Several algorithms for the solution of problems involving the planning of transmission systems have been
proposed (see [1] for more details). In this letter, a specialized
GA that is a metaheuristic is presented to solve the static and
multistage TEP problem. Known systems were used to show
the applicability of this tool. A comparison was made among
the proposed GA and other metaheuristics.
T
II. EFFICIENT GENETIC ALGORITHM
The presented GA is based on [3] (initially designed to solve
the generalized assignment problem) with some modifications
that were added to solve the TEP problem. The proposed GA
has some special characteristics such as: 1) it uses fitness and
unfitness functions to identify the value of the objective function and the unfeasibility, respectively, of the tested solution; 2)
it applies an efficient strategy of local improvement for each individual tested; and 3) it substitutes only one individual in the
population for each iteration.
For the TEP problem, the fitness represents the total costs of
planned lines to be constructed and the individual unfeasibility
(unfitness) represents the total sum of load shedding caused by
an individual. In most GAs, the unfeasibilities are penalized in
Manuscript received August 29, 2006; revised April 16, 2007. This work was
supported by the Brazilian institutions CNPq and FAPESP. Paper no. PESL00063-2006.
R. Romero is with the Faculty of Engineering of Ilha Solteira, Paulista State
University, Ilha Solteira, SP, Brazil (e-mail: ruben@dee.feis.unesp.br).
M. J. Rider and I. de J. Silva are with the Department of Electric Energy
Systems, University of Campinas, Campinas, SP, Brazil (e-mail: mjrider@dsee.
fee.unicamp.br; irenio@dsee.fee.unicamp.br).
Digital Object Identifier 10.1109/TPWRS.2007.907592
Fig. 1. GA flowchart.
the objective function or the unfeasible solutions are discarded.
The GA presented in [3] does not require regulation of the parameter used to penalize the unfeasibilities of the algorithms that
use only one fitness function. The fitness function is used to implement the selection and in the substitution of an individual
in the population when all members of the population are feasible. The unfitness function is used to substitute an individual
in the population when there are proposals of unfeasible solutions in the current population. The presented GA substitutes, in
each step, only one individual of the population. The offspring
is incorporated into the population according to the following
procedure: 1) if the offspring is unfeasible, it can only replace a
more unfeasible offspring; 2) if the offspring is feasible, then it
must substitute the most unfeasible individual from the current
population. If all members of the current population are feasible,
then the offspring must substitute the individual that presents a
greater investment value; and 3) the offspring must be different
from all the members of the population. If it is equal to a member
in the population, it must be discarded.
The previous proposal presents very simple conceptual
changes to the traditional GA: 1) all saved solutions in the
current population are different; this prevents the premature
convergence that is very common in conventional GAs; 2) the
local improvement phase allows a more efficient evolution of
the GA; and 3) the substitution logic in the current population
preserves the best created topologies, and the topologies are
only discarded when better offspring are created. This strategy
is better than the elitism proposal. The process stops if the best
solution found does not improve after a specified number of
iterations. The GA flowchart is shown in Fig. 1.
A. Modifications
The presented GA suggests modifying the algorithm shown
in [3] in the three topics: (1) in the local improvement of the
generated offspring phase; (2) in the generation of the initial
population; and (3) in the increase of diversity control.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
TABLE I
RESULTS FOR STATIC TEP PROBLEM
The local improvement of an individual is one of the main
contributions of the proposed GA. If the created offspring is unfeasible (it has load shedding), then the unfeasibility should be
improved using the constructive heuristic algorithm (CHA) proposed by [4]. Considering that the main objective of this phase is
to completely eliminate individual unfeasibility, the CHA will
add lines to the individuals in order to eliminate the unfeasibility. Some lines of the individual are unnecessary and should
be discarded so that the individual (solution proposal) does not
become too expensive. Thus, an ordering of all the solution proposal lines in decreasing order of costs is carried out, and the
removal, one by one, of all the lines, is carried out. Those that,
when the removal of the individual is simulated, do not present
load shedding are the unnecessary lines and are therefore discarded. The lines that remain are those that present load shedding when their removal is simulated. Considering a randomly
generated population, the GA may need a higher computational
effort, especially in medium and large systems. In this letter, the
population initialization suggested in [6] is implemented. In the
creation of offspring, a selection by tournaments is used; recombination of 100%, the mutation, and the decimal codification
are made in accordance with [6] and [2] for static and multistage TEP problems, respectively. The modifications made in
the presented GA improve the search process while initializing
and creating only new feasible solutions, thus creating a search
process that is different to the GA presented in [3]. The control
of the diversity can be extended, and due to this fact, an offspring
can enter the current population: 1) if it presents better quality
than the worst quality stored in the current population, and 2) if
it is different to each one of the elements of the current population in a minimum number of elements of the coded vector.
III. TESTS AND RESULTS
The GA proposed to solve the static TEP problem was tested
using: the Garver system, the IEEE 24-bus system, and the
South Brazilian system. The electrical data can be found in [1]
and [5]. The results are shown in Table I. To estimate the computational effort of the GA, a comparison was made among the
average solutions for the linear programming problems (LPs)
carried out by GA with the results presented in the literature.
From the summary presented in Table II (results from over 50
TABLE II
COMPUTATIONAL EFFORT
trials, used the same for all cases), it can be noticed that the
GA presented shows better performance than the other metaheuristics tested in the TEP problem because it executes fewer
LPs to find the optimal solution for the tested systems. The
solved LPs are equivalent for all the metaheuristics presented in
Table II. The GA was tested to also solve the multistage TEP.
The algorithm was tested in the Colombian system. The same
criteria used in [2] were used here. Thus, the transmission lines
appear in the objective function with their
chosen in stage
appear multiplied by a
nominal values; the ones added in
and those added in
appear multiplied by
factor
to consider the annual discount rate in the
a factor
multistage TEP, as proposed in [2]. In [2], this system, which
was solved using a specialized GA, found the following results:
;
Total investment:
Plan P1 (
US$):
–
–
Plan P2
–
–
–
–
US$):
(
–
–
–
Plan P3 (
US$):
–
–
–
–
–
–
–
.
–
–
–
The specialized GA proposed in [2] solved on average
210 000 LPs to reach its optimal solution. The GA presented
in this letter found the following results: Total investment:
Plan P1 (
US$):
–
–
–
–
–
Plan P2 (
US$):
–
–
Plan
P3 (
US$):
–
–
–
–
–
–
–
.
–
–
Thus, the presented GA showed itself to be superior to the algorithm presented in [2], leading to a better quality investment
proposal. It must also be observed that the topologies found are
significantly different, that is, even though the number of paths
used in both topologies is significantly small (20 paths), their
expansion proposals are different in 11 (over 50%) of the used
paths. The presented algorithm converges after solving around
20 500 to 64 000 LPs. Thus, the presented algorithm also converges solving a smaller number of LPs.
IV. CONCLUSION
An efficient genetic algorithm was applied to solve the static
and multistage TEP. The achieved results for medium and large
systems show the excellent performance of this GA.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
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The proposed GA presented better performance than the other
metaheuristics to solve the static and multistage TEP problem,
solving fewer LPs to find the optimal solutions. It has also found
a better solution for the multistage TEP problem when compared to the GA proposed in [2] (8.6 million US$ cheaper).
[2] A. H. Escobar, R. A. Gallego, and R. Romero, “Multistage and coordinated planning of the expansion of transmission systems,” IEEE Trans.
Power Syst., vol. 19, no. 2, pp. 735–744, May 2004.
[3] P. C. Chu and J. E. Beasley, “A genetic algorithm for the generalized
assignment problem,” Comput. Oper. Res., vol. 24, no. 1, pp. 17–23,
1997.
[4] R. Villasana, L. L. Garver, and S. J. Salon, “Transmission network planning using linear programming,” IEEE Trans. Power App. Syst., vol.
PAS-104, no. 2, pp. 349–356, Feb. 1985.
[5] R. Fang and D. J. Hill, “A new strategy for transmission expansion in
competitive electricity markets,” IEEE Trans. Power Syst., vol. 18, no.
1, pp. 374–380, Feb. 2003.
[6] R. Gallego, “Long term transmission systems planning using combinatorial optimization,” (in Portuguese) Ph.D. dissertation, State Univ.
Campinas, Campinas, Brazil, 1997.
REFERENCES
[1] R. Romero, A. Monticelli, A. Garcia, and S. Haffner, “Test systems and
mathematical models for transmission network expansion planning,”
Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 149, no. 1, pp.
29–36, Jan. 2002.
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