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An Initialization Procedure in Solving Optimal Power Flow by Genetic Algorithm Mirko Todorovski and Dragoslav Rajiˇcic´, 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. Rajiˇcic´ 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
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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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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)
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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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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
ˇ ´ : INITIALIZATION PROCEDURE IN SOLVING OPF BY GA TODOROVSKI AND RAJICIC
TABLE III TIME CONSUMPTIONS USING DIFFERENT POWER FLOW METHODS
TABLE IV RESULTS WHEN CHROMOSOMES INCLUDE GENERATOR REAL POWERS INSTEAD OF VOLTAGE ANGLES
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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.
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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
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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.
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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 Rajiˇcic´ (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.”
