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Ant Colony System Algorithm for the Planning of Primary Distribution Circuits J. F. Gómez, H. M. Khodr, Member, IEEE, P. M. De Oliveira, L. Ocque, J. M. Yusta, Member, IEEE, R. Villasana, Senior Member, IEEE, and A. J. Urdaneta, Senior Member, IEEE
Abstract—The planning problem of electrical power distribution networks, stated as a mixed nonlinear integer optimization problem, is solved using the ant colony system algorithm (ACS). The behavior of real ants has inspired the development of the ACS algorithm, an improved version of the ant system (AS) algorithm, which reproduces the technique used by ants to construct their food recollection routes from their nest, and where a set of artificial ants cooperate to find the best solution through the interchange of the information contained in the pheromone deposits of the different trajectories. This metaheuristic approach has proven to be very robust when applied to global optimization problems of a combinatorial nature, such as the traveling salesman and the quadratic assignment problem, and is favorably compared to other solution approaches such as genetic algorithms (GAs) and simulated annealing techniques. In this work, the ACS methodology is coupled with a conventional distribution system load-flow algorithm and adapted to solve the primary distribution system planning problem. The application of the proposed methodology to two real cases is presented: a 34.5-kV system with 23 nodes from the oil industry and a more complex 10-kV electrical distribution system with 201 nodes that feeds an urban area. The performance of the proposed approach outstands positively when compared to GAs, obtaining improved results with significant reductions in the solution time. The technique is shown as a flexible and powerful tool for the distribution system planning engineers. Index Terms—Circuit optimization, optimization methods, power distribution planning.
NOMENCLATURE Conductor type subindex. Substation size index. Set of conductor types. Set of substation sizes. Fixed cost coefficient of a proposed line in route . Fixed cost coefficient of substation at node . Variable cost coefficient of a line in route . Variable-cost coefficient of substation at node . Maximum apparent power demand at node . Subindex for “existent” circuit or substation. Subindex for “proposed” circuit or substation. Set of routes associated to existent lines. Set of proposed routes for the construction of future lines. Route subindex between nodes and . Manuscript received July 7, 2003. J. F. Gómez, H. M. Khodr, P. M. De Oliveira, L. Ocque, R. Villasana, and A. J. Urdaneta are with the Universidad Simón Bolívar, Caracas 1062, Venezuela (e-mail:
[email protected]). J. M. Yusta is with the Universidad de Zaragoza, Zaragoza 50015, Spain. Digital Object Identifier 10.1109/TPWRS.2004.825867
,
Set of frontier nodes of the growing grid. Set of nodes left to be fed, with a proposed path directly connected to the frontier nodes of the growing grid. Pheromone accumulation factor. Arc length (in kilometers). Loss factor: average losses between maximum losses. Objective function value of the best network that has been found at that point of the algorithm. Objective function value of grid constructed by ant colony . Number of ant colonies. Total number of nodes associated to the distribution system. Set of nodes associated to existing substations. Set of nodes associated to proposed substations. Probability that an ant colony , located at a set of chooses to move to node . frontier nodes Random number with uniform probability distribution between zero and one (0-1). Real variable that determines the relative importance of the exploitation over the exploration . Random variable defined by probability distribution . Apparent power (in kilovolt-amperes) transported by route . Apparent power (in kilovolt-amperes) provided by substation . Maximum capacity (in kilovolt-amperes) of circuit in route . Maximum capacity (in kilovolt-amperes) of substation located at node . Calculated voltage at node (in kilovolt-amperes). Nominal voltage. Binary decision variable associated to the installation of a circuit of size in route . Binary decision variable associated to the installation of a substation of size in route . Decay rate of the pheromone level due to evaporation. Parameter that determines the relative importance of the pheromone level and the heuristic search . function Maximum and minimum allowed operation voltages.
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Heuristic search function Pheromone level. I. INTRODUCTION
T
HIS paper is organized in six sections: containing an introduction, the statement of the planning problem of primary distribution circuits, the description of the proposed ant colony system methodology with emphasis in the differences with the ant system algorithm, the description of the results obtained with the application of the methodology to a test case, and its comparison with the results obtained by a genetic-algorithm (GA) approach, and the conclusions derived from the developed work. Projecting a distribution network minimizing the installation and operation costs is a complicated task, due to the high number of technically feasible alternatives, leading to the use of mathematical optimization tools, which could result in important savings for the electric utilities due to the magnitude of the required investments. The selected project must satisfy the electric demands with acceptable reliability levels, at a minimum cost, taking into account the power transportation limits of distribution lines and substations, the maximum and minimum allowed voltage levels at the nodes of the grid, and the radial structure of the network during operation [1]–[4]. In general, the planning problem of distribution networks may be stated as an optimization problem, so that for a given geographical area or region with a set of load demands previously estimated, the characteristics of the network are determined, including the location of the transformation centers, required for the load supply, minimizing the total installation and operation costs, subject to the technical requirements for a satisfactory operation of the system, such as voltage levels and transportation capacities of the elements of the system. A complete survey of the proposed techniques for the solution of the planning problem of primary distribution circuits can be found in [5] and [6]. Initially, the proposed methods were mainly based upon the generation and evaluation of possible solutions, oriented to small size problems, and requiring important efforts for the production of the alternatives to be evaluated. Among these the heuristic zone valuation and the generation of service areas methods may be mentioned. They rely completely upon the experience of the planning engineer and have the disadvantage that the best alternative may not be considered. The planning problem of primary distribution networks has been basically stated as a classical mixed integer linear programming problem, where an objective function that includes both the investment and the operation costs of the network, is minimized subject to technical constraints related with the characteristics of the electric service [7]. This formulation includes binary variables linked to the fixed costs associated to some of the decision variables as well as linear approximations to represent the variable costs. Stated as such, the classical branch-and-bound techniques [7] have a natural application. Although the continuous research
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and improvement of these techniques keeps them as an alternative always to be considered, the linear approximations as well as the combinatorial complexity of the problem question their application to electric systems of real dimensions due to the requirements of excessive computing resources. This is the main reason for the application of alternative approaches, such as those classified as metaheuristic methods that are able to locate good solutions with reduced computational effort. In order to reduce the computation requirements of the branch-and-bound optimization technique, the identification of the set of variables that affect most of the results was proposed in order to reduce the complexity of the mathematical model [6]; however, a linear approximation of the variable costs is required and excessive solution times are reported as the number of binary variables increases. Heuristic search methods have been developed [7], [8], showing faster performance than the conventional optimization techniques but with some limitations in the goodness of the solutions to the problem that are obtained. In [10], an open-loop planning procedure is proposed where an expert system based upon the use of geographical information is applied to automate the selection of the primary and secondary circuit routes. GAs have also been applied to the solution of this problem, reducing the solution time and enhancing the obtained results [5], [8], [11]–[13]. In [12], a solution methodology based upon a GA is proposed for the design of primary distribution circuits considering multiple system expansion stages, where a special coding procedure is proposed to reduce the number of unfeasible solutions evaluated by the algorithm. These methodologies have been improved significantly. An evolutionary programming algorithm is presented in [15], considering the uncertainties of the possible scenarios. In [6] and [14], the potential of the GAs is shown in comparison with classical optimization techniques, to solve the planning problem in a very complete and detailed formulation, considering the nonlinearity of the cost function, the limits on the voltage magnitudes, and a term in the objective function to take into account the reliability of the system, reporting significant improvements in the solution times. An integer variable coding scheme was used to facilitate the consideration of different conductor sizes and substation sizes; also, new genetic operators were proposed to improve the performance of the algorithm. In [6], the approach is expanded to consider multiple development stages as well as multiple objectives. In [21], an evolutionary approach is applied to the design of a medium voltage network of a real city, using a very detailed model of the network. Ant system (AS) algorithms have been applied successfully to different optimization problems such as the traveling salesman and the quadratic assignment problems [16]. Ants are members of a family of social insects, which live in organized colonies. Some are capable of finding the way in very complex mazes and use this ability to establish food recollection roots from the nest. Even though their individual learning capabilities are very limited, the complexity of the organization of the colony allows a very efficient communication, based upon tactile and chemical media. During their expeditions in search of food sources, they
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liberate chemical secretions or pheromones to mark the paths that have been used, guiding in this way, the new explorers to the food sources. The shortest paths will tend to have a higher magnitude of the pheromone deposits and, therefore, will be preferred by the explorers. The ant colony system (ACS) algorithm is an improvement of the AS algorithm–more robust, faster, and with a better probability of achieving the global solution; its performance for the solution of classical optimization problems is favorably compared with GAs and simulated annealing techniques [17]. In this work, the ACS algorithm is adapted to the solution of the optimal planning problem of primary distribution circuits for a given maximum load condition, using a distribution system ac load-flow solution module. The technique is applied to two test cases taken from the literature [6], [14], [18]. The article is devoted to the solution of the single-stage planning problem, where an existent network is expanded in a planning period, although the long-term dynamics of the loads due to changes in the land use, to the effect of population fluctuations, as well as the changes in the available technology, among other factors may also be considered by the proposed methodology. The results reflect an interesting potential of this metaheuristic approach and encourage further research on the topic. II. STATEMENT OF THE PROBLEM The problem may be stated as an optimization problem as follows: Minimize an objective function representing the fixed costs correspondent to the investment in lines and substations and the variable costs associated to the operation of the system, expressed by the following equation:
(1) Subject to the following constraints: A) Energy balance constraint at all the system nodes (2) B)
Capacities of the distribution circuits i) Proposed circuits (3) ii) Existing circuits
C)
ii) Existent substations: (6) D)
Limits on the node voltage magnitudes The distribution system load-flow algorithm proposed in [19] as presented in [1] was used for the evaluation of the behavior of the network; however, it is clear that other techniques could be used instead. Once the node voltages have been calculated, the following expression is used for the evaluation of the correspondent constraint: (7)
E)
Radial network restriction The radial characteristic of the network is enforced by the proposed branch selection approach described in Section III-E-III. III. PROPOSED METHODOLOGY
The three layer mini-max solution scheme extended for the consideration of multiple perturbations illustrated by Fig. 1 was applied [20], where system experts select a number of relevant scenarios or perturbations at the top layer. These selected scenarios are used at the intermediate layer or optimization level, for the statement of the optimization problem, which is solved using the output information of the lower layer or simulation level, consisting of a distribution system load-flow program. The ACS algorithm was adapted for the specific problem and used at the optimization layer. A. ACS Algorithm The ACS algorithm represents an improvement with respect to the AS or ant colony algorithms, which are a type of black-box optimization approaches that aim to maximize an objective function, subject to a set of nonlinear constraints [16]. Ant colony algorithms or AS are based in the behavior of these insects with exceptional abilities to find the shortest paths to their food sources without visual help, using a chemical substance called pheromone that is deposited as they walk, to mark their trailways as they walk. Initially, a group of individuals explores the surface without a predetermined direction. After food is found, the individuals go back to the colony. As all of the individuals travel approximately at the same speed, the shortest paths have a tendency to contain a higher level of pheromone because more individuals have used them. After a short time period, the differences among the pheromone deposits in the routes are big enough to influence the decision of the new individuals, which will decide toward the shortest paths, producing a feedback to the system that contributes and promotes the use of the best paths.
(4)
B. Mathematical Formulation of the ACS Algorithm
(5)
AS basically use two functions to guide the search toward the optimal solution of the problem. 1) a function which is proportional to the amount of pheromone deposited;
Capacities of the substations i) Proposed substations
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The pheromone plays the role of the long-term memory, of the ant colonies, being distributed in the different paths of the network, allowing indirect communication between the different ant colonies. C. Advances of the ACS With Respect to the AS
Fig. 1. Proposed methodology.
2) a heuristic guide function , also referred to as the incremental cost function and generally defined as the inverse of the distance, which constitutes an auxiliary function that helps to generate better grids. The proposed algorithm works as follows: initially, a random level of pheromone is deposited in each branch of the initial grid. Then, a number of ants accomplishes independent explorations through the different branches of the grid, guided by the heuristic guide function and by the amount of pheromone deposited in the branches, according to probabilistic transition rules, until all of the load nodes are fed, completing an expedition of that ant colony. The probabilistic transition rule used in the ant systems [16], [17], is called proportional random rule, given by (11), defines located at a set of frontier the probability that ants of colony choose to move toward node nodes if in any other case.
(9) After a predetermined number of sets of independent expeditions or ant colonies are performed, the pheromone levels are updated. A local pheromone revision rule is implemented to vary the pheromone deposits. A fraction of the pheromone evaporates, and it is assumed that each ant colony contributes to increase the level of pheromone of the transited paths in proportion to the total cost of the resultant grid. AS use also a global pheromone revision rule to simulate the effect of the addition of new deposits by the ant colonies that transit through the net, as well as the evaporation phenomena; the pheromone of each of the paths is updated considering the grids that were constructed by all of the ant colonies, as follows is [16]: (see (10) and (11) at the bottom of the page) where the total cost of the grid constructed by ant colony .
The ACS incorporates three main differences with respect to the AS algorithm: i) A new proportional pseudo-random transition rule, that weights the priorities of the exploration of new paths with the use of the accumulated knowledge of the problem, in order to improve the selection of the “best route.” In the ACS algorithm, an ant positioned at node , selects the branch to transit to node according to the following rule: (see (12) at the bottom of the page). When a particular ant is positioned in node , a random . If , then number is generated the best branch is selected, this means that exploitation was the decisive factor, while in the opposite case, the selection of the route is performed according to the probabilistic transition rule given by (11). ii) The global pheromone level revision rule is applied only to those branches that belong to the best networks found so far. Pheromone is deposited only in those branches that belong to the best network. This change aims to make a more direct search, orienting the explorations toward the best one found so far. This modification is somehow similar to the elite strategies of the genetic algorithms or evolutionary programming approaches. The pheromone level is updated as follows: (13) if best network found otherwise. (14) iii) A local pheromone level revision rule in which these levels are updated during the route generation process (15) where
is the minimum pheromone level.
(10) if grid found by ant colony in any other case.
if otherwise.
(11)
(12)
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In [16], three different approaches were tested for this rule: Q-learning, Ant-Q, and this one, reporting better results for the latter two approaches, recommending the application of the rule described by (17), due to its simplicity. D. Simplified Flowchart of the ACS Algorithm A simplified flowchart of the algorithm is presented in Fig. 2 to illustrate the proposed methodology. E. Application of the ACS Algorithm to the Planning of Distribution Systems The application of the ACS algorithm to the planning of distribution systems requires the definition of the objective function to be maximized, as well as a heuristic guide function. 1) Objective Function: The objective function of the problem was defined as the inverse of the total costs, as defined in Section II, considering the equipment costs and the operation losses required for each alternative, which are calculated with the available information from the load flow. 2) Heuristic Guide Function: The heuristic guide function plays a preponderant role during the first stages of the optimization process due to the fact that it allows the generation of low-cost networks with a good voltage behavior. However, as the algorithm evolves, the level of pheromone accumulated in the branches rises, and the decision of which branch to take is less dependent on the heuristic function. In the AS algorithm, the heuristic guide function is defined as the inverse of the length, giving in this way preference to the selection of the shortest paths. In this particular application of the ACS algorithm to the distribution system planning problem, the following variables were included in the heuristic guide function: the length, the incremental cost of the network, and the magnitude of the load at the end of the path. It was defined as the weighted sum of two terms. The first term is related with the kilovolt-amperes (kVA) times the length, given in (kVA.m) and the second term with the incremental cost of adding the new branch to the grid, given in monetary units per kVA fed ($/kVA). A comparative study was conducted to evaluate the use of this function. Four cases were analyzed: 1) without the heuristic guide function; 2) only with the first term (kVA.m); 3) only with the second term (incremental cost); 4) using both terms. After 1000 runs of each alternative, the results of this comparison show the benefits of the use of the complete function, with a higher probability of reaching better solutions in a reduced computer time. The probability of reaching the global optimum in 30 sets of expeditions was 0.07, 0.83, 1.00, and 1.00 for the four alternatives, respectively; the number of different simulated cases was 574, 223, 211, and 192, the number of solutions within 1% of tolerance of the objective function was 36, 82, 120, and 99, and the expected number of required expeditions to reach the optimum was 30, 15, 6.7, and 6.3, respectively. 3) Branch Selection: The branch selection process of the ACS methodology was adapted to the distribution system planning problem. In particular, for the selection of every new
Fig. 2. Simplified flowchart of the algorithm.
branch to be added to the network, an evaluation is performed of all the branches that are connected from the set frontier nodes of the growing grid to nodes that are left to be fed. F. Tuning of the Parameters of the Algorithm A statistical study was performed on the basis of 1000 runs of the algorithm, in order to identify the best range of values for the parameters of the heuristic function and for the parameters that control the functioning of the algorithm. It must be said, however, that the algorithm is very robust, and that the variations of the parameters mostly affect the characteristics of the convergence: (number of expeditions i) Number of ant colonies per set) This number was varied between 4 and 50. The amount of different simulated cases increases from 62 to 347, the expected number of required sets of expeditions to reach the optimum decreases gradually from 12 (for ten ant colonies) to 4.3 and the number of solutions close to the optimum increases to 13. It was found that for the 23-node system, the impact over the performance indexes is reduced gradually as the number of ant colonies increases to more than 20. This number is related to the complexity of the search to be performed and, therefore, to the number of nodes of the distribution system. for each ii) The pheromone accumulation factor expedition, was varied in the range (10–100). It was shown that the performance of the algorithm does not vary substantially. However, for a value of 1000, the expected number of required expeditions increases drastically to 12.
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TABLE I COORDINATES OF THE LOADS
Fig. 3.
Proposed feasible routes. TABLE II ECONOMIC DATA
iii) Maximum pheromone level of the first set of expeditions. Was varied between 1 and 100, obtaining an almost null influence on the algorithm performance. iv) Exploitation versus exploration weighting factor . Was varied between 0.1 and 1.0. The convergence of the algorithm is accelerated, but over 0.6 the probability of the reaching the global optimum is drastically decreased. The number of simulated cases is also reduced (21 cases for 0.1). was varied from v) The pheromone evaporation factor 0.2 to 1. It was found that the number of different simulated cases and the number of solutions close to the optimum decreased (from 183 to 121 and from 11 to 9, respectively), deteriorating the probability of reaching the global optimum, while the reduction in the expected number of required sets of expeditions (from 6.5 to 5.1) accelerating the convergence of the method. : varied from 1 to vi) Heuristic function weighting factor 4, the number of different simulated cases was reduced from 183 to 53, as well as the number of solutions close to the optimum (from 11 to 9), suggesting the use of the value of 1. vii) Maximum number of sets of expeditions. A value of higher than 30 sets was required to obtain a probability of 1.0 of reaching the global optimum for sets consisting of 20 ACs. G. Calculation of the Cost of the Energy Losses The power flow through each circuit for the maximum load condition is calculated using a well-known distribution load-flow algorithm [19] as presented in [1]. The power losses
in the grid are calculated using the load-flow results for the maximum load condition. Then, the energy losses for the period of one year are calculated multiplying the power losses and by for the maximum load condition by the loss factor the number of hours in one year (8760). The associated cost of the energy losses is calculated according to the costs of the energy in ($/kW/year). The present value of these yearly costs is calculated according to the discount rate and to the time period under study. IV. RESULTS A. Test Case 1 A 34.5-kV distribution system with a 10-MVA substation to feed an oil production area was chosen as test case 1. The maximum allowed voltage deviation is 3%. The average power factor is equal to 0.9. The capacity factor is equal to one and the proposed conductor sizes for the distribution lines are 1/0 and 4/0 in simple or double circuit. Table I presents the plane coordinates of the substation, and of the load nodes, corresponding to the real physical locations of the wells and other oil production facilities. The proposed feasible routes, defined according to the geography and the topology, are shown in Fig. 3. The economic data required for the analysis is presented in Table II. B. ACS Results After six (6) sets of expeditions, the proposed algorithm arrived at the network presented in Fig. 4. The results are summarized in Table III. The resultant grid is similar, although slightly better, than the network presented in [18] using a GA. The differences in the
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Fig. 5.
Fig. 4.
Optimal network found by the ACS algorithm.
TABLE III ACS ALGORITHM RESULTS
Cost of the network versus number of expeditions.
C. Test Case 2 A more complex 10-kV large distribution system with a 10-MVA substation to feed an urban area was chosen as test case 2 [6], [14]. The characteristics of the system are given in [6] and are not presented here due to space limitations. The solution presented in [6] and [14] was obtained by means of an elaborated, high-performance GA, after 1200 generations of 150 individuals (around 200 000 cases evaluated). The average performance on test case 2 for 20 runs of the ACS algorithm with a set of 150 ACs is illustrated by Fig. 6. It can be observed that the algorithm required 156 sets of expeditions (23 400 evaluations) to reach the 0.5% threshold, 392 sets of expeditions (58 800 evaluations) for the 0.2%, and 562 expeditions (84 300 evaluations) to reach the 0.1% threshold, requiring 26 min, 1 h, 5 min, and 1 h, 33 min, respectively, to reach the solution in a personal computer (PC) with an 800-MHz processor with 128-MB random-access memory (RAM) in a Microsoft Windows 98 environment. One of the resultant grids is presented in Fig. 7. As many metaheuristic approaches, the algorithm showed to be very efficient at the beginning of the search, obtaining important decreases in the cost of the grid in very few iterations; however, the more precise search required after a number of trials has a slower performance. D. Improvements to the ACS Algorithm
total cost of the circuits are in the order of 0.03%. This fact represents an improvement of the ACS algorithm with respect to the GA, especially considering that the computational effort is considerably reduced. The algorithm was run 1000 times in order to evaluate its performance. Results are presented in Fig. 5, where the minimum, average, and maximum costs are presented versus the number of expeditions.
Three modifications to the ACS algorithm were proposed and tested on test case one. Although they arrived at the same solution, their statistical performance was independently evaluated, performing 30 sets of expeditions for the test case: i) The magnitude of the pheromone accumulated in each expedition was set as proportional to the stage of the process. This change decreased the convergence time and the expected number of sets of expeditions required to reach the global optimum from 6.3 to 5.7, but reduced the probability of achieving the global optimum as well (from 1.0 to 0.985), reducing the expected number of simulated cases from 192 to 154, narrowing the search space. The expected number of solutions within 1% of
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iii) The replacement of already evaluated solutions was evaluated. If after an expedition, the obtained solution has already been evaluated, then a new attempt is performed. The replacement is performed in a finite number of trials. After a sensitivity study, it was determined that the best number of maximum replacement attempts was ten (10), improving slightly both the convergence of the algorithm and the number of solutions close to the optimum (from 12 to 14). It also increased in 100% the number of different cases that were simulated (from 192 to 598), and slightly decreasing the expected number of required sets of expeditions to 6.1. V. CONCLUSION
Fig. 6. ACS average performance in test case 2.
A new methodology, based upon the ACS algorithm, is proposed for the planning of electric energy distribution systems. The methodology is very flexible and calculates the location and the characteristics of the circuits minimizing the investment and operation costs while enforcing the technical constraints such as the transmission capabilities and the limits on the voltage magnitudes, allowing the consideration of a very complete and detailed model for the electric system. In particular, a single objective mixed nonlinear-integer programming model was used for the application of the methodology. The application of the methodology on two test cases: a 23-node and a 201-node distribution system, showed the feasibility of the application of the proposed methodology, presenting a significant reduction of the computational effort required when compared to those obtained by a high-performance GA. The technique is shown as a flexible and powerful tool for the distribution system planning engineers. The results encourage the use and further development of the methodology. ACKNOWLEDGMENT The authors are grateful to J. L. Bernal-Agustín from Centro Politécnico Superior of Universidad de Zaragoza, Spain, for the assistance provided with the second test case. REFERENCES
Fig. 7.
Resultant grid for test case 2.
the optimum objective function value also decreased from 12 to 11. ii) Introduction of two different types of ants. The second type of ants decides which route to take, considering only the original pheromone level, without weighting it with the heuristic function. This change pursued the increase of the number of different cases to be evaluated (it increased from 192 to 320). A large increase in the convergence time was introduced, escalating the expected number of required sets expeditions from 6.3 to 12.2, but decreasing the expected number of good solutions (within 1% of objective function tolerance) 12 to 11, and the probability of achieving the global optimum to 0.98. This change was discarded.
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[10] Z. Sumic et al., “Automated underground residential distribution design–Part 2: Prototype implementation and results,” IEEE Trans. Power Delivery, vol. 8, pp. 644–650, Apr. 1993. [11] E.-C. Yeh, S. S. Venkata, and Z. Sumic, “Improved distribution system planning using computational evolution,” IEEE Trans. Power Syst., vol. 11, pp. 668–674, May 1996. [12] V. Miranda, J. V. Ranito, and L. M. Proença, “Genetic algorithm in optimal multistage distribution network planning,” IEEE Trans. Power Syst., vol. 9, pp. 1927–1933, Nov. 1994. [13] I. J. Ramirez-Rosado and J. L. Bernal-Agustín, “Optimization of power distribution network design by application of genetic algorithm,” Int. J. Power Energy Syst., vol. 15, no. 3, pp. 104–110, 1995. , “Genetic algorithm applied to design of large power distribution [14] system,” IEEE Trans. Power Syst., vol. 13, pp. 696–703, May 1998. [15] P. M. S. Carvalho and L. A. F. M. Ferreira, “Optimal distribution network expansion planning under uncertainty by evolutionary decision convergence,” Int. J. Elect. Power Energy Syst., vol. 20, no. 2, pp. 125–129, 1998. [16] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: Optimization by a colony of cooperating agents,” IEEE Trans. Syst., Man, Cybern. B, vol. 26, pp. 29–41, Feb. 1996. [17] M. Dorigo and L. M. Gambardella, “Ant colony system: A cooperative learning approach to the traveling salesman problem,” IEEE Trans. Evol. Comput., vol. 1, pp. 29–41, Apr. 1997. [18] Y. Da Silva, J. Di Girolano, and A. Ferreira, “Optimización de circuitos aéreos de distribución en campos petroleros” (in Spanish), in Proc. I IEEE Andean Region Conf., vol. II, 1999, pp. 967–972. [19] C. E. Advirson, “Diversified demand method of estimation residential distribution transformer load,” Edison Elect. Inst. Bulletin, vol. 8, pp. 469–479, Oct. 1940. [20] A. J. Urdaneta and V. Chankong, “A multiple objective minimax approach for controller setting of systems running under disturbances,” in Control: Theory and Advanced Technology, Invited Paper Special Issue on Multiple Objective, Discrete, Dynamic Systems. Tokyo, Japan: MITA, 1989, vol. 5. [21] E. Diaz-Dorado, J. Cidrás, and E. Miguéz, “Application of evolutionary algorithms for the planning of urban distribution networks of medium voltage,” IEEE Trans. Power Syst., vol. 17, pp. 879–884, Aug. 2002.
J. F. Gómez received the electrical engineering and M.Sc. degrees from Universidad Simón Bolívar (USB), Caracas, Venezuela, in 1997 and 2000, respectively. Currently, he is an Assistant Professor of Electrical Engineering in the Department of Energy Conversion and Delivery at USB. He has participated in many projects for the local industry. His research interests include the planning and optimization of electrical power systems.
H. M. Khodr (M’99) received the B.Sc.-M.Sc. and the Ph.D. degrees in electrical engineering from the José Antonio Echeverría Higher Polytechnic Institute in 1993 and 1997, respectively. Currently, he is an Associate Professor of Electrical Engineering in the Department of Energy Conversion and Delivery at Universidad Simón Bolívar, Caracas, Venezuela. He has been responsible for a number of projects performed for local industries. His current research activities include planning, operation, and economics of electrical distribution and industrial power systems.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 2, MAY 2004
P. M. De Oliveira received the electrical engineering and M.Sc. degrees from Universidad Simón Bolívar (USB), Caracas, Venezuela, in 1995 and 2002, respectively. Currently, he is an Assistant Professor of Electrical Engineering at USB. He has participated in different projects in the area of inspection, basic, and detailed engineering for the oil industry, electric utilities, and telecommunication companies.
L. Ocque received the electrical engineering degree from Instituto Universitario de las Fuerzas Armadas Nacionales (IUPFAN) in 1997 and the M.Sc. degree in electrical engineering from Universidad Simón Bolívar (USB), Caracas, Venezuela, in 2003. Currently, he is a Project Engineer with the Instituto de Energía (INDENEUSB). He was with the Engineering Department at C.V.G. Electrificación del Caroní (EDELCA), Caracas, Venezuela.
J. M. Yusta (M’01) received the industrial engineer degree in 1994 from the Engineering Higher Polytechnical Center of the Universidad de Zaragoza, Zaragoza, Spain, in 1994, and the Ph.D. degree from the Universidad de Zaragoza in 2000. Currently, he is an Associate Professor of Electrical Engineering at the Universidad de Zaragoza. His research interests include technical and economical problems of electrical distribution systems.
R. Villasana (SM’97) received the electrical engineer degree from the Universidad de Oriente, Barcelona, in 1971, the M.Sc. degree in electrical engineering from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1973, and the Ph.D. degree in electrical engineering from Renseelaer Polytechnic Institute, Troy, NY, in 1984. Currently, he is a Professor of Electrical Engineering in the Department of Energy Conversion and Delivery at Universidad Simón Bolívar, Caracas, Venezuela. His research interests include power system planning and operation. Dr. Villasana is Former Dean of Professional Studies and Former Vice Chairman of the IEEE Venezuelan Section.
A. J. Urdaneta (SM’90) received the electrical engineer degree (Hons.) from Universidad Simón Bolívar (USB), Caracas, Venezuela, in 1979. He received the M.Sc. degree in electrical engineering and applied physics and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH, in 1983 and 1986, respectively. He has been responsible for a number of projects and studies performed for local industries. His research interests include power system analysis and optimization. Currently, he is Professor of Electrical Engineering in the Department of Energy Conversion and Delivery at USB. Dr. Urdaneta is Former Dean of Professional Studies and Former Chairman of the IEEE Venezuelan Section.
