Optimal Power Flow by Enhanced Genetic Algorithm

Transcription

1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY Optimal Power Flow by Enhanced Genetic Algorithm Anastasios G. Bakirtzis, Senior Member, IEEE, Pandel N. Biskas, Student Member, IEEE, Christoforos E. Zoumas, Student Member, IEEE, and Vasilios Petridis, Member, IEEE Abstract This paper presents an enhanced genetic algorithm (EGA) for the solution of the optimal power flow (OPF) with both continuous and discrete control variables. The continuous control variables modeled are unit active power outputs and generator-bus voltage magnitudes, while the discrete ones are transformer-tap settings and switchable shunt devices. A number of functional operating constraints, such as branch flow limits, load bus voltage magnitude limits, and generator reactive capabilities, are included as penalties in the GA fitness function (FF). Advanced and problem-specific operators are introduced in order to enhance the algorithm s efficiency and accuracy. Numerical results on two test systems are presented and compared with results of other approaches. Index Terms Genetic algorithms (GAs), optimal power flow (OPF). NOMENCLATURE Bus voltage angle vector. Load (PQ) bus voltage magnitude vector. Unit active power output vector. Generation (PV) bus voltage magnitude vector. Transformer tap settings vector. Bus shunt admittance vector. System state vector. System control vector. A hat above vectors and denotes that the entry corresponding to the slack bus is missing. For simplicity of notation, it is assumed that there is only one generating unit connected on a bus. This assumption is relaxed in SectionV. I. INTRODUCTION SINCE its introduction as network constrained economic dispatch by Carpentier [1] and its definition as optimal power flow (OPF) by Dommel and Tinney [2], the OPF problem has been the subject of intensive research. The OPF optimizes a power system operating objective function (such as the operating cost of thermal resources) while satisfying a set of system operating constraints, including constraints dictated by the electric network. OPF has been widely used in power system operation and planning [3]. After the electricity sector restructuring, OPF has been used to assess the spatial variation of electricity prices and as a congestion management and pricing tool [4]. In its most general formulation, the OPF is a nonlinear, nonconvex, large-scale, static optimization problem with both Manuscript received October 9, 2000; revised August 20, The authors are with the Department of Electrical Engineering, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece. Publisher Item Identifier S (02) continuous and discrete control variables. Even in the absence of nonconvex unit operating cost functions, unit prohibited operating zones, and discrete control variables, the OPF problem is nonconvex due to the existence of the nonlinear (AC) power flow equality constraints. The presence of discrete control variables, such as switchable shunt devices, transformer tap positions, and phase shifters, further complicates the problem solution. The literature on OPF is vast, and [5] presents the major contributions in this area. Mathematical programming approaches, such as nonlinear programming (NLP) [6] [9], quadratic programming (QP) [10], [11], and linear programming (LP) [12] [14], have been used for the solution of the OPF problem. Some methods, instead of solving the original problem, solve the problem s Karush Kuhn Tucker (KKT) optimality conditions. For equality-constrained optimization problems, the KKT conditions are a set of nonlinear equations, which can be solved using a Newton-type algorithm. In Newton OPF [15], the inequality constraints are added as quadratic penalty terms to the problem objective, multiplied by appropriate penalty multipliers. Interior point (IP) methods [16] [18], convert the inequality constraints to equalities by the introduction of nonnegative slack variables. A logarithmic barrier function of the slack variables is then added to the objective function, multiplied by a barrier parameter, which is gradually reduced to zero during the solution process. The unlimited point algorithm [19] uses a transformation of the slack and dual variables of the inequality constraints which converts the OPF problem KKT conditions to a set of nonlinear equations, thus avoiding the heuristic rules for barrier parameter reduction required by IP methods. OPF programs based on mathematical programming approaches are used daily to solve very large OPF problems. However, they are not guaranteed to converge to the global optimum of the general nonconvex OPF problem, although there exists some empirical evidence on the uniqueness of the OPF solution within the domain of interest [20]. To avoid the prohibitive computational requirements of mixed-integer programming, discrete control variables are initially treated as continuous, and post-processing discretization logic is subsequently applied [21], [22]. Whereas the effects of discretization on load tap changing transformers are small and usually negligible [20], the rounding of switchable shunt devices may lead to voltage infeasibility, especially when the discrete VAR steps are large, and requires special logic [22]. The handling of nonconvex OPF objective functions, as well as the unit prohibited operating zones, also present problems to mathematical programming OPF approaches /02$ IEEE

2 230 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002 Recent attempts to overcome the limitations of the mathematical programming approaches include the application of simulated annealing-type methods [23], [24], and genetic algorithms (GAs) [25], [26]. In [25], a simple genetic algorithm (SGA) is used for OPF solution. The control variables modeled are generator active power outputs and voltages, shunt devices, and transformer taps. Branch flow, reactive generation, and voltage magnitude constraints are treated as quadratic penalty terms in the GA fitness function (FF). To keep the GA chromosome size small, only a 4-bit chromosome area is used for the encoding of each control variable. A sequential GA solution scheme is employed to achieve acceptable control variable resolution. Test results on the IEEE 30-bus system, comprising 25 control variables, are presented. In [26], a GA is used to solve the optimal power dispatch problem for a multinode auction market. The GA maximizes the total participants welfare, subject to network flow and transport limitation constraints. The nodal real and reactive power injections that clear the market are selected as the problem control variables. A GA with two advanced operators, namely, elitism and hill climbing, is used. A 10-bit chromosome area is devoted to each control variable. Test results on a 17-node, 34-control variable system are presented. The GA-OPF approaches overcome the limitations of the conventional approaches in the modeling of nonconvex cost functions, discrete control variables, and prohibited unit operating zones. However, they do not scale easily to larger problems, since the solution deteriorates with the increase of the chromosome length, i.e., the number of control variables. Thus, the test results in the existing GA-OPF literature are limited to very small problems. This paper presents an enhanced genetic algorithm (EGA) for the solution of the OPF. The control variables and constraints included in the OPF and the penalty method treatment of the functional operating constraints are similar to the ones in [25] with the following improvements: switchable shunt devices and transformer taps are modeled as discrete control variables. Variable binary string length is used for different types of control variables, so as to achieve the desired resolution for each type of control variable, without unnecessarily increasing the size of the GA chromosome. In addition to the basic genetic operators of the SGA used in [25] and the advanced ones used in [26], problem-specific operators, inspired by the nature of the OPF problem, have been incorporated in our EGA. With the incorporation of the problem-specific operators, the GA can solve larger OPF problems. Test results on systems with up to 242 buses and 500 control variables demonstrate the improvement achieved with the aid of problem-specific operators. II. OPTIMAL POWER FLOW PROBLEM FORMULATION The OPF problem can be formulated as a mathematical optimization problem as follows: Min (1) S.t. (2) (3) (4) where The equality constraints (2) are the nonlinear power flow equations. The inequality constraints (3) are the functional operating constraints, such as branch flow limits (MVA, MW or A); load bus voltage magnitude limits; generator reactive capabilities; slack bus active power output limits. Constraints (4) define the feasibility region of the problem control variables such as unit active power output limits; generation bus voltage magnitude limits; transformer-tap setting limits (discrete values); bus shunt admittance limits (continuous or discrete control). III. GENETIC ALGORITHMS GAs are general purpose optimization algorithms based on the mechanics of natural selection and genetics. They operate on string structures (chromosomes), typically a concatenated list of binary digits representing a coding of the control parameters (phenotype) of a given problem. Chromosomes themselves are composed of genes. The real value of a control parameter, encoded in a gene, is called an allele [27]. GAs are an attractive alternative to other optimization methods because of their robustness. There are three major differences between GAs and conventional optimization algorithms. First, GAs operate on the encoded string of the problem parameters rather than the actual parameters of the problem. Each string can be thought of as a chromosome that completely describes one candidate solution to the problem. Second, GAs use a population of points rather than a single point in their search. This allows the GA to explore several areas of the search space simultaneously, reducing the probability of finding local optima. Third, GAs do not require any prior knowledge, space limitations, or special properties of the function to be optimized, such as smoothness, convexity, unimodality, or existence of derivatives. They only require the evaluation of the so-called fitness function (FF) to assign a quality value to every solution produced. Assuming an initial random population produced and evaluated, genetic evolution takes place by means of three basic genetic operators: 1) parent selection; 2) crossover; 3) mutation. Parent selection is a simple procedure whereby two chromosomes are selected from the parent population based on their fitness value. Solutions with high fitness values have a high probability of contributing new offspring to the next generation. The selection rule used in our approach is a simple roulette-wheel selection [27]. (5) (6)

3 BAKIRTZIS et al.: OPTIMAL POWER FLOW BY ENHANCED GENETIC ALGORITHM 231 Fig. 2. GA chromosome structure. Fig. 1. Simple genetic algorithm (SGA). Crossover is an extremely important operator for the GA. It is responsible for the structure recombination (information exchange between mating chromosomes) and the convergence speed of the GA and is usually applied with high probability ( ). The chromosomes of the two parents selected are combined to form new chromosomes that inherit segments of information stored in parent chromosomes. Until now, many crossover schemes, such as single point, multipoint, or uniform crossover have been proposed in the literature. Uniform crossover [28] has been used in our implementation. While crossover is the main genetic operator exploiting the information included in the current generation, it does not produce new information. Mutation is the operator responsible for the injection of new information. With a small probability, random bits of the offspring chromosomes flip from 0 to 1 and vice versa and give new characteristics that do not exist in the parent population [27]. In our approach, the mutation operator is applied with a relatively small probability ( ) to every bit of the chromosome. The FF evaluation and genetic evolution take part in an iterative procedure, which ends when a maximum number of generations is reached, as shown in Fig. 1. When applying GAs to solve a particular optimization problem (OPF in our case), two main issues must be addressed: 1) the encoding, i.e., how the problem physical decision variables are translated to a GA chromosome and its inverse operator, decoding; 2) the definition of the FF to be maximized by the GA (the GA FF is formed by an appropriate transformation of the initial problem objective function augmented by penalty terms that penalize the violation of the problem constraints [29]). IV. GENETIC ALGORITHM SOLUTION TO OPTIMAL POWER FLOW A. Encoding The chromosome is formed as shown in Fig. 2. There are four chromosome regions (one for each set of control variables), namely, 1) ; 2) ; 3) ; and 4). Encoding is performed using different gene-lengths for each set of control variables, depending on the desired accuracy. The decoding of a chromosome to the problem physical variables is performed as follows: 1) continuous controls taking values in the interval and 2) discrete controls taking values with (7) (8) where is the decimal number to which the binary number in a gene is decoded and is the gene length (number of bits) used for encoding control variable. B. Fitness Function (FF) GAs are usually designed so as to maximize the FF, which is a measure of the quality of each candidate solution. The objective of the OPF problem is to minimize the total operating cost (1). Therefore, a transformation is needed to convert the cost objective of the OPF problem to an appropriate FF to be maximized by the GA. The OPF functional operating constraints (3) are included in the GA solution by augmenting the GA FF by appropriate penalty terms for each violated functional constraint. Constraints on the control variables (4) are automatically satisfied by the selected GA encoding/decoding scheme (7) and (8). Therefore, the GA FF is formed as follows: (9) (10)

4 232 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002 where fitness function; constant; fuel cost function of unit (in our case, a quadratic function); weighting factor of functional operating constraint ; penalty function for functional operating constraint ; violation of th functional operating constraint, if positive; Heaviside (step) function; number of units; number of functional operating constraints. Given a candidate solution to the problem, represented by a chromosome, the FF is computed as follows. Step 1) Decode the chromosome to determine the actual control variables,, using (7) and (8). The computed control vector satisfies, by design, constraints (4). Step 2) Solve the power flow (2) to compute the state vector,. Step 3) Determine the violated functional constraints (3) and compute associated penalty functions (10). Step 4) Compute the FF using (9). In Step 2, a simple fast decoupled load flow (FDLF) [30] is used with no PV-PQ bus-type switching, since generator reactive capabilities are incorporated in the functional operating constraints and no local control adjustments, such as tap and switchable shunts [31], since the settings of these controls are determined by the GA. Therefore, only a few load flow iterations are required for convergence. The FDLF and matrices are formed and factorized only once in the beginning the effect of the changes of shunt admittances on the matrix is neglected. In case that, due to the random (yet within limits) initial selection of the control variables, the load flow does not converge within a predefined number of iterations (set to 8), large penalty terms, proportional to the maximum active/reactive power mismatch, are added to the FF. C. Advanced and Problem-Specific Genetic Operators One of the most important issues in the genetic evolution is the effective rearrangement of the genotype information. In the SGA crossover is the main genetic operator responsible for the exploitation of information while mutation brings new nonexistent bit structures. It is widely recognized that the SGA scheme is capable of locating the neighborhood of the optimal or near-optimal solutions, but, in general, requires a large number of generations to converge. This problem becomes more intense for large-scale optimization problems with difficult search spaces and lengthy chromosomes, where the possibility for the SGA to get trapped in local optima increases and the convergence speed of the SGA decreases. At this point, a suitable combination of the basic, advanced, and problem-specific genetic operators must be introduced in order to enhance the performance of the GA. Advanced Fig. 3. Gene swap operator. and problem-specific genetic operators usually combine local search techniques and expertise derived from the nature of the problem. A set of advanced and problem-specific genetic operators has been added to the SGA in order to increase its convergence speed and improve the quality of solutions. Our interest was focused on constructing simple yet powerful enhanced genetic operators that effectively explore the problem search space. The advanced features included in our GA implementation are as follows. 1) Fitness Scaling: In order to avoid early domination of extraordinary strings and to encourage a healthy competition among equals, a scaling of the fitness of the population is necessary [27]. In our approach, the fitness is scaled by a linear transformation. 2) Elitism: Elitism ensures that the best solution found thus far is never lost when moving from one generation to another. The best solution of each generation replaces a randomly selected chromosome in the new generation [32]. 3) Hill Climbing: In order to increase the GA search speed at smooth areas of the search space a hill-climbing operator is introduced, which perturbs a randomly selected control variable. The modified chromosome is accepted if there is an increase in FF value; otherwise, the old chromosome remains unchanged. This operator is applied only to the best chromosome (elite) of every generation [26], [29]. In addition to the above advanced features, which are called advanced despite their wide use in most recent GA implementations to distinguish between the SGA and our EGA, operators specific to the OPF problem have been added. All problem-specific operators introduce random modification to all chromosomes of a new generation. If the modified chromosome proves to have better fitness, it replaces the original one in the new population. Otherwise, the original chromosome is retained in the new population. All problem-specific operators are applied with a probability of 0.2. The following problem-specific operators have been used. 1) Gene Swap Operator (GSO): This operator randomly selects two genes in a chromosome and swaps their values, as shown in Fig. 3. This operator swaps the active power output of two units, the voltage magnitude of two generation buses, etc. Swapping among different types of control variables is not allowed. 2) Gene Cross-Swap Operator (GCSO): The GCSO is a variant of the GSO. It randomly selects two different chromosomes from the population and two genes, one from every selected chromosome, and swaps their values, as shown in Fig. 4. While crossover exchanges information between high-fit chromosomes, the GCSO searches for alternative alleles, exploiting information stored even in low-fit strings.

5 BAKIRTZIS et al.: OPTIMAL POWER FLOW BY ENHANCED GENETIC ALGORITHM 233 Fig. 4. Gene cross-swap operator. Fig. 5. Gene copy operator. Fig. 8. Enhanced genetic algorithm (EGA). Fig. 6. Fig. 7. Gene inverse operator. Gene max-min operator. 3) Gene Copy Operator (GCO): This operator randomly selects one gene in a chromosome and with equal probability copies its value to the predecessor or the successor gene of the same control type, as shown in Fig. 5. This operator has been introduced in order to force consecutive controls (e.g., identical units on the same bus) to operate at the same output level. 4) Gene Inverse Operator (GIO): This operator acts like a sophisticated mutation operator. It randomly selects one gene in a chromosome and inverses its bit-values from one to zero and vice versa, as shown in Fig. 6. The GIO searches for bit-structures of improved performance, exploits new areas of the search space far away from the current solution, and retains the diversity of the population. 5) Gene Max-Min Operator (GMMO): The GMMO tries to identify binding control variable upper/lower limit constraints. It selects a random gene in a chromosome and, with the same probability (0.5), fills its area with 1 s or 0 s, as shown in Fig. 7. D. Enhanced Genetic Algorithm (EGA) In the EGA, shown in Fig. 8, after the application of the basic genetic operators (parent selection, crossover, and mutation) the advanced and problem-specific operators are applied to produce the new generation. All chromosomes in the initial population are created at random (every bit in the chromosome has equal probability of being switched ON or OFF). Due to the decoding process selected [(7) and (8)], the corresponding control variables of the initial population satisfy their upper lower bound or discrete value constraints (4). However, the initial population candidate solutions may not satisfy the functional operating constraints (3) or even the load flow constraints (2) since the random, within limits, selection of the control variables may lead to load flow divergence (as already discussed in Section I V-B). Population statistics computed for the new generation include maximum, minimum, and average fitness values and the 90% percentile. Population statistics are then used to adaptively change the crossover and mutation probabilities [33]. If premature convergence is detected the mutation probability is increased and the crossover probability is decreased. The contrary happens in the case of high population diversity. V. TEST RESULTS In this section, the proposed EGA solution of the OPF is evaluated using two test systems: 1) the IEEE 30-bus system [6] and 2) the 3-area IEEE RTS96 [34]. The test examples have been run on a 1.4-GHz Pentium-IV PC. Twenty runs have been performed for each case examined. The results which follow are the best solution over these 20 runs. A. IEEE 30-Bus System The first test system is the IEEE 30-bus, 41-branch system [6]. It has a total of 24 control variables as follows: five unit active power outputs, six generator-bus voltage magnitudes, four transformer-tap settings, and nine bus shunt admittances. The gene length for unit power outputs is 12 bits and for generator voltage magnitudes is 8 bits. They are both treated as continuous controls. The transformer-tap settings can take 17 discrete values (each one is encoded using 5 bits): the lower and

6 234 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 2, MAY 2002 TABLE I IEEE 30-BUS SYSTEM RESULTS Fig. 9. FF comparison for IEEE 30-bus system. upper limits are 0.9 p.u. and 1.1 p.u., respectively, and the step size is p.u. The bus shunt admittances can take six discrete values (each one is encoded using 3 bits): the lower and upper limits are 0.0 p.u. and 0.05 p.u., respectively, and the step is 0.01 p.u. (on system MVA basis). The GA population size is taken equal to 80, the maximum number of generations is 200, and crossover and mutation are applied with initial probability 0.9 and 0.001, respectively. Two sets of 20 test runs were performed; the first (SGA) with only the basic GA operators and the second (EGA) with all operators, including advanced and problem-specific operators. The FF evolution of the best of these runs is shown in Fig. 9. The best and worst solutions of the second set of 20 runs (EGA) are shown in Table I. The operating costs of the best and worst solutions are $/h and $/h, respectively, (0.01% difference). The differences between the values of the control variables in the best and worst solutions are not significant. The operating cost of all EGA-OPF solutions is slightly less than the $/h figure reported in [6]. As shown in Table I, there is a slight difference in unit marginal costs (UMCs), attributed to network losses. Note that, in this case, the UMCs coincide with the nodal prices, since no unit limits are reached. Fig. 9 demonstrates the improvement achieved with the inclusion of the advanced and problem-specific operators. The SGA run took 18 s, while the EGA took 76 s to evaluate 200 generations. However, the EGA provides a far better solution than SGA even in the first 25 generations, or 10 s. B. IEEE 3-Area RTS96 The 3-area IEEE RTS-96 [34] is a 73-bus, 120-branch system. It consists of three areas connected through five tie lines. The area-a unit cost data are derived from the heat rate data provided in [34] and the fuel cost data listed in Table II. The value of water is zero, assuming excessive inflows. Area-B and area-c fuel costs are selected three times the area-a fuel costs, to impose exports from area A to areas B and C. A contingency case with tie lines and out of service, under 90% peak load conditions, is studied. To impose congestion, the ratings of tie lines and are reduced by 50% (to 250 MVA). This system has a total of 150 control variables as follows: 98 unit active power outputs, 33 generator-bus voltage magnitudes, 16 transformer tap-settings, and 3 bus shunt admittances. TABLE II FUEL COSTS FOR IEEE 3-AREA RTS-96 The lower and upper limits of voltage magnitude of all buses are 0.95 p.u. and 1.05 p.u., respectively, (except for PV buses where p.u.). Transformer taps take discrete values within 0.9 p.u. and 1.1 p.u. with a step size of p.u (17 discrete values). Similarly, bus shunt admittances take discrete values between 150 MVAR (inductor, at rated voltage) and 0 MVAR with a 50 MVAR step (four discrete values). The GA population size is taken equal to 180, the maximum number of generations is 600, and crossover and mutation are applied with initial probability 0.9 and 0.001, respectively. It was necessary to increase both the population size and the maximum number of generations to solve the larger problem. It was also necessary to increase the probability of application of problem-specific operators from 0.2 to 0.5. First, the unconstrained schedule is obtained by ignoring branch flow limits. Branch flow limits are ignored by selecting the corresponding penalty weight to zero in (9). The unconstrained schedule results in an 81.8 MVA overloading of tie line The corresponding operating cost is $/h. Next, the constrained schedule is calculated by activating the branch flow constraints. Tie line flow is now reduced to MVA (almost to the 250 MVA line rating). The operating cost is increased to $/h due to congestion. The FF evolution of both the SGA and the EGA, shown in Fig. 10, demonstrates the improvement achieved with the inclusion of the advanced and the problem-specific operators. The SGA run took 266 s, while the EGA took 1643 s to evaluate 600 generations. However, as shown in Fig. 10, a far better solution is provided by EGA during the same execution time as SGA.

7 BAKIRTZIS et al.: OPTIMAL POWER FLOW BY ENHANCED GENETIC ALGORITHM 235 TABLE III COMPUTATIONAL REQUIREMENTS Fig. 10. FF comparison for IEEE 3-area RTS96. VI. COMPUTATIONAL REQUIREMENTS OF GENETIC ALGORITHM OPTIMAL POWER FLOW In an SGA, if a population of size PS is allowed to evolve for a total number of NG generations, the product PG NG determines the required number of FF evaluations (NFE) and hence the GA computational requirements. When problem-specific operators are used, the required number of fitness evaluations increases accordingly. In GA-OPF, the FF evaluation is synonymous with power flow solution in terms of computational requirements, since the latter is the computationally dominant task in the FF evaluation procedure (see Section IV-B). It is widely recognized among GA practitioners that the required NFE for a particular GA implementation depends on problem difficulty, which, in turn, depends on two factors: 1) the chromosome length and 2) the shape and characteristics of the fitness landscape. Problems with smooth fitness landscapes are easy to solve with GA. If the global optimum is located at the bottom of a steep gorge of the fitness landscape, the GA may require a large number of fitness evaluations to locate it. Thus, two optimization problems with the same chromosome length may require vastly different NFE to solve owing to the difference in their fitness landscapes. In GA-OPF, the chromosome length is determined by the number of control variables and the resolution required for each control type. The number of buses affects the fitness evaluation ( power flow solution) time. The fitness landscape of the GA-OPF is very hard to visualize, except for trivial problems employing at most two-decision variables. For the assessment of the GA-OPF computational requirements, an experiment is designed as follows. Four test systems of increasing size are created, based on the IEEE RTS96 [34] (1-, 3-, 5-, and 10-area configurations). The GA population size is 200, the probability of application of problem-specific operators is 0.5, and the maximum number of generations is 600 in all four cases. Table III summarizes the results of 20 test runs in all test systems. The last four columns report the average (over the 20 runs) computational requirements of the GA. The number of generations (NG) to arrive at a good quality OPF solution is reported in the 7th row. A good quality OPF solution is one with fitness value within 0.1% of the fitness obtained after allowing the GA to evolve for 600 generations (well within the flat portion of Fig. 10). The execution time to arrive at a good quality solution is reported in the 9th row. The results of Table III show that the difference of the best and worst solutions increases slightly and the execution time increases considerably as the system size increases. VII. CONCLUSIONS A GA solution to the OPF problem has been presented and applied to small and medium size power systems. The main advantage of the GA solution to the OPF problem is its modeling flexibility: nonconvex unit cost functions, prohibited unit operating zones, discrete control variables, and complex, nonlinear constraints can be easily modeled. Another advantage is that it can be easily coded to work on parallel computers. The main disadvantage of GAs is that they are stochastic algorithms and the solution they provide to the OPF problem is not guaranteed to be the optimum. Another disadvantage is that the execution time and the quality of the provided solution deteriorate with the increase of the chromosome length, i.e., the OPF problem size. 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Eng. degree from the National Technical University of Athens, Athens, Greece, in 1979, and the M.S.E.E. and Ph.D. degrees from Georgia Institute of Technology, Atlanta, in 1981 and 1984, respectively. In 1984, he was a consultant to Southern Company, Atlanta, GA. Since 1986, he has been with the Department of Electrical Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece, where he is currently an Associate Professor. His research interests are in power system operation and control, reliability analysis, and alternative energy sources. Pandel N. Biskas (S 01) received the Dipl. Electr. Eng. degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1999, where he is currently pursuing the Ph.D. degree. His research interests are in power system operation and control and transmission pricing. Christoforos E. Zoumas (S 98) received the Dipl. Electr. Eng. degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996, where he is currently pursuing the Ph.D. degree. His research interest is in computer applications in power systems. Vasilios Petridis (M 77) received the B.S. degree in electrical engineering from the National Technical University, Athens, Greece, in 1969, and the M.Sc. and Ph.D. degrees in electronics and systems from King s College, University of London, London, U.K., in 1970 and 1974, respectively. He has been Consultant of the Naval Research Centre, Greece, and Director of the Department of Electronics and Computer Engineering and Vice-Chairman of the Faculty of Electrical and Computer Engineering at Aristotle University, Thessaloniki, Greece. He is currently Professor in the Department of Electronics and Computer Engineering in the Aristotle University of Thessaloniki, Thessaloniki, Greece. He is coauthor of the monograph Predictive Modular Neural Networks: Application to Time Series (Norwell, MA: Kluwer, 1998). He is also author of four books on control and measurement systems and approximately 110 research papers. His research interests include control systems, machine learning, intelligent and autonomous systems, artificial neural networks, evolutionary algorithms, fuzzy systems, modeling and identification, robotics, and industrial automation.