Network-Constrained Economic Dispatch Using Real-Coded Genetic Algorithm

Transcription

1 198 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003 Network-Constrained Economic Dispatch Using Real-Coded Genetic Algorithm Ioannis G. Damousis, Student Member, IEEE, Anastasios G. Bakirtzis, Senior Member, IEEE, and Petros S. Dokopoulos, Member, IEEE Abstract A genetic algorithm (GA) solution to the networkconstrained economic dispatch problem is presented. A real-coded GA has been implemented to minimize the dispatch cost while satisfying generating unit and branch power-flow limits. A binarycoded GA was also developed to provide a means of comparison. GA solutions do not impose any convexity restrictions on the dispatch problem. The proposed method was applied on the electrical grid of Crete Island with satisfactory results. Various tests with convex and nonconvex unit cost functions demonstrate that the proposed GA locates the optimum solution, while it is more efficient than the binary-coded GA. Index Terms Economic dispatch, genetic algorithm (GA). I. INTRODUCTION THE network-constrained Economic Dispatch (NC-ED) schedules the outputs of the online generating units so as to meet the system demand at minimum cost while satisfying transmission line power-flow limits. NC-ED produces a feasible economic schedule that takes into account the transmission network limitations. In addition, various contingencies and scenarios can be investigated in order to improve the security and reliability of the system. The NC-ED is a generalization of the classical economic dispatch (ED) problem [1], [28], which ignores transmission network limitations. The NC-ED can also be viewed as a special case of the OPF problem [2] in which the objective is the fuel cost minimization and the nonlinear (ac) power-flow equations are approximated by the linear (dc) power-flow equations. Thus, the NC-ED is also named dc optimal power flow (DC-OPF). The three power system optimization problems are classied in decreasing degree of generality and complexity as follows: OPF, NC-ED, and ED. All three problems are in general nonconvex optimization problems. Nonconvexities in the general OPF problem arise from a) nonconvex unit cost functions, due to valve points or combined cycle units [3]; b) nonconvex feasible set of unit active-power outputs, due to zones of prohibited operation; c) nonlinear power-flow equality constraints; Manuscript received March 21, 2001; revised January 29, This work was supported by the European Union Research Project More Advanced Control Advice for Secure Operation of Isolated Power Systems with Increased Renewable Energy Penetration and Storage, Contract ERK5-CT The authors are with the Electrical Power Systems Laboratory, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, , Greece. Digital Object Identier /TPWRS d) discrete control variables, such as switched shunt devices, transformer taps, and phase shters. In the classical ED and the NC-ED, only the two first sources of nonconvexities apply (however, when transmission losses are modeled by quadratic loss formula, the third source of nonconvexity also applies in the form of a single nonlinear power balance equality constraint). The classical example of optimization problem, solved by standard nonlinear programming techniques, is that of minimizing a convex function over a convex set [4]. This convex minimization problem is guaranteed to have a unique local minimum, which is also the global minimum. Convex minimization problems can be solved using conventional gradient, subgradient, or Newton-based local search techniques, exploiting the fact that local extrema are also global. By using modeling simplications, convex ED and NC-ED problems are efficiently solved using traditional, local search ED algorithms such as lambda iteration (which ignores network constraints) [1], [28], [3], linear programming [5], and quadratic programming [6]. However, for the reasons stated above, both the ED and the NC-ED are in general nonconvex optimization problems. Nonconvex optimization problems have, in general, many local minima, and local search techniques cannot locate the global minimum, since they may be trapped in a local minimum. Recently, new advances in the field of multiextemal global optimization have been used to solve many problems in science and engineering. There are three broad categories of global optimization approaches [7]: a) deterministic approaches; b) stochastic approaches; and c) metaheuristics. The deterministic approaches [8] exploit analytical properties of the problem to generate a deterministic sequence of points converging to a global optimal solution. Two analytical properties, convexity and monotonicity, have been most successfully exploited, giving rise to two important trends dc optimization (dealing with problems described by means of dferences of convex functions or sets) and monotonic optimization (dealing with problems described by means of functions monotonically increasing or decreasing along rays). One of the most intensely studied problems of global optimization is the linearly constrained concave minimization problem, which seeks to minimize a concave function over a polyhedron,. Studied for more than three decades, the methods used for its solution have been refined and extended to more general dc optimization problems. It is solved using the outer approximation method [8], which constructs a nested sequence of polyhedra such that is obtained from by imposing an additional /03$ IEEE

2 DAMOUSIS et al.: NETWORK-CONSTRAINED ECONOMIC DISPATCH USING REAL-CODED GENETIC ALGORITHM 199 linear constraint (cutting plane) chosen so as to ensure that the minimum of the objective function over monotonically converges (from below) to the minimum of the objective function over. Stochastic global optimization approaches [9] require little or no additional assumptions (such as convexity) on the optimization problem, at the expense of at most being able to provide a probabilistic convergence guarantee. The three main classes of stochastic methods are [7] two-phase methods [10], random search methods [11], and random function methods [12]. Metaheuristics (or modern heuristics) are methods that are often based on processes observed in physics or biology. They require no additional assumptions on the optimization problem and they are classied as heuristics since, with the exception of simulated annealing [13], no formal proof on their convergence to the global optimum exists. Metaheuristics have been successfully applied to the solution of the ED problem. The four main classes of metaheuristics are: simulated annealing [14], tabu search [15], GAs [16], and evolutionary programming [17]. GAs, in particular, have been successfully employed in order to overcome the nonconvexity problems of the conventional ED algorithms (valve points, [18], [19], and prohibited operating zones [20]). Also, GA s ability to model almost any kind of constraints in the form of penalty functions or by using various chromosome coding schemes tailored to the specic problem [21] makes it an attractive method for the solution of the ED problem. The main disadvantages of GAs over traditional methods are a) their long execution time; and b) the fact that they are not guaranteed to converge to the global optimal solution. This paper extends previous work on GA solution to the ED problem in two ways: the proposed GA approach a) is based on a floating-point coding scheme, which improves accuracy and reduces the execution time and b) observes transmission line power-flow limits. For comparison purposes, a classic binary-coded GA scheme was also implemented. Several test cases, on a real, small-sized power system, were investigated in order to demonstrate the robustness and the generic nature of the proposed method. For the verication of the GA results, the lambda iteration and the quadratic programming method were used as benchmarks for the ED and the NC-ED problems, respectively. II. NOTATION number of buses; bus index; number of branches; branch index; number of units; unit index; unit active-power output vector (size ); bus-to-unit incidence matrix (size ); active-power output of the th unit in megawatts; th unit cost function in dollars per hour; feasible operating region of unit ; it includes unit operating limits ( ) [possibly adjusted for regulation, spinning reserve, or ramping] and may also include unit prohibited operating zones; bus active-power demand vector (size ); negative network admittance matrix (size ); bus voltage phase-angle vector (size ); a constant network matrix defining branch power flows as a function of the bus voltage phase angles; computed as the inverse of the branch impedance matrix times the node to branch incidence matrix (size ); vector of branch active power-flow limits (size ). It is assumed that the last ( th) unit is the reference unit and it is connected to the last ( th) bus, so that is the active-power output of the reference unit and is the voltage phase angle of the slack (reference) bus. A tilde ( ) above a vector (matrix) denotes that the vector entry (matrix row and column) corresponding to the system slack bus or reference unit is omitted. For example, is the reduced negative network admittance matrix, with slack bus excluded. III. NETWORK-CONSTRAINED ECONOMIC DISPATCH MATHEMATICAL FORMULATION The network-constrained economic dispatch (NC-ED) or dc optimal power flow (DC-OPF) is formulated as an optimization problem as follows: subject to (1) (2) (3) (4) where objective (1) is the total system operating cost to be minimized; (2) represents the system dc power-flow equations; (3) represents branch power-flow limits (normal or emergency, depending on system conditions); and (4) define the feasible region of operation of the generating units. The dc power-flow equations (2), representing the activepower balance at every node of the system, are linearly dependent (det ). Thus, the voltage phase angle of a slack or reference bus can be arbitrarily chosen ( ), while the active-power output of a reference unit must satisfy the system power balance equation, which is derived by adding all dc (lossless) power-flow equations (2) together where is the total system power demand,. The remaining set of ( ) power-flow equations can be solved for the ( ) unknown voltage phase angles, given the ( ) unit active-power outputs and the ( ) nodal demands. (5) (6)

3 200 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003 As discussed in the introduction, problem (1) (4) is in general a nonlinear, nonconvex optimization problem and conventional (local search) optimization methods cannot locate its global optimum solution. A real-coded GA is described in the next section for the general NC-ED problem solution. Two special NC-ED cases, amenable to solution by conventional optimization methods, are described below and used as benchmarks in the results section. Convex Network-Constrained Economic Dispatch: For the special case that the cost function is a convex function, for example, a convex quadratic function (7) and (4) are limited to generator minimum maximum operating limits the NC-ED problem (1), (7), (2), (3), and (8) is a convex optimization problem and can be solved, for example, as a quadratic programming (QP) problem [6]. Classic Economic Dispatch Problem: If the branch power-flow limits (3) are neglected, the dc power-flow equations (2) can be replaced by a single system power balance equation (5). The optimization problem (1), (5), and (8) is the classic ED problem (losses neglected). It can be solved using the -iteration method [3]. IV. GA SOLUTION GAs are inspired by the study of genetics [22], [23]. They are conceptually based on natural evolution mechanisms working on populations of solutions in contrast to other search techniques that work on a single solution. The most interesting aspect of GAs is that although they do not require any prior knowledge or space limitations, such as smoothness or convexity of the function to be optimized, they exhibit very good performance on the majority of the problems applied. Initially, GAs were designed to work on a bit representation of the problem parameters [16]. In recent studies though, the superiority of higher cardinality alphabet GAs (floating point or integer) was demonstrated with their application on various problems [25]. A description of a real-coded GA developed for the solution of the NC-ED problem is given next, followed by a brief description of a binary-coded GA developed for comparison purposes. A. Real-Coded GA In the GA solution, the outputs of the ( ) free units can be chosen arbitrarily within limits (4) while the output of the reference unit is constrained by the system power balance equation (5). A floating-point number represents the output of each free generator. The outputs of the generators are concatenated to form a consolidated solution string of ( ) numbers called genotype or chromosome. An obvious advantage over binarycoded GAs is that with the use of floating-point numbers, we (8) achieve the absolute precision overcoming the critical decision of how many bits should be used for the encoding of each parameter. A population of genotypes is initially generated at random. The population size is an important parameter of GA and is selected after experimentation with the specic problem. The initial population is evolved through a chromosome evaluation and an evolution process described next. 1) Chromosome Evaluation: Given a chromosome, its fitness function is calculated in the following steps. Step 1: Chromosome decomposition. From the definition of the genotype given before, it is obvious that the genotype coincides with the reduced unit power-output vector, which contains the outputs of the free units. Given, the output of the reference unit is computed using (5). Step 2: The total operating cost is computed as the sum of the individual unit costs Step 3: The bus voltage phase-angle vector is next computed using (6). Using (3), the power flow of each branch is calculated. Step 4: To account for reference unit limit violations (4) and branch power-flow limit violations (3) the total operating cost is augmented by nonnegative penalty terms and, respectively, penalizing constraint violations. Thus, the augmented cost function is formed (9) (10) The penalty terms and are proportional to the corresponding violations and zero in case of no violation. They are chosen high enough as to make constraint violations prohibitive in the final solution and where (11) (12) (13) and is the Heaviside (unit step) function. Step 5: The GA fitness function is computed as the inverse of the augmented cost function (14)

4 DAMOUSIS et al.: NETWORK-CONSTRAINED ECONOMIC DISPATCH USING REAL-CODED GENETIC ALGORITHM 201 where is a constant. The constant is used in order to prevent the fitness function from obtaining too small values and its magnitude should be of the order of the system maximum operating cost. 2) Population Evolution: The evolution of the population takes place following the general GA principles through selection, crossover, and mutation [16]. Selection: After the evaluation of the initial randomly generated population, the GA begins the creation of the new generation. Chromosomes from the parent population are selected in pairs with a probability proportional to their fitness to replicate and form offspring chromosomes. This selection scheme is known as Roulette wheel selection [16]. After the selection and a probability test [16] is passed, the parent chromosomes are combined and mutated in order to form the offspring chromosomes. Since the encoding of the parameters is not the classical binary one, new crossover and mutation operators must be used. Crossover: the -arithmetical crossover operator was used [26]. If chromosomes, and from generation are to be crossed, four possible children are created with with for (15) Parameter is a constant equal to 0.3 for our experiments. The two children that have the higher fitness are chosen to replace the parents in the new population. Mutation: Every parameter of the offspring chromosomes undergoes a probability test and it is passed [16], the mutation operator alters that parameter using Michalewicz s nonunorm mutation operator [24]. This operator is described below. If is a power output chromosome vector and is th unit s output that is chosen to be mutated, the new output will be after mutation (16) where is a random bit, and function returns a value in the range [0, ] such that the probability of the value returned being close to 0 increases with (17) where is a random floating-point number in the interval [0, 1]; is the current generation; is the maximum number of generations; and is a parameter that determines the degree of dependence on the number of generations. In this way, the operator makes a unorm search at the beginning of the evolution and in later stages narrows the search around the local area of the parameter resembling a hill-climbing operator. For our experiments, was chosen equal to 5 [27]. Elitism: The previous procedure described for the two chromosomes is repeated until all of the chromosomes of the parent generation are replaced by the newly formed chromosomes. The best chromosome of the parent generation and the best chromosome found in all of the previous generations are copied intact to the next generation, so that the possibility of their destruction through a genetic operator is eliminated. According to the schemata theory [16], the new generation usually provides a better average fitness. 3) GA-II: An Enhancement of the Real-Coded GA: The evaluation of network constraint violations using (6) and (3) in step 3 of the chromosome evaluation is the most computationally demanding task of the evolution process. In order to reduce the execution time, a variant of the fitness evaluation is tested. Specically, only the chromosomes that present operating cost (9) less than 110% of the parent population s minimum operating cost go through the time-demanding network-constraints violation calculation (step 3). For the rest of the chromosomes that represent a relatively bad solution, the calculation of (6) and (3) is omitted and a large penalty term is added to their augmented cost function (10) so that they have small possibility of being selected as parents for the next generation. We name this variation of the real-coded GA, GA-II. B. Binary-Coded GA For comparison purposes, a standard binary-coded GA was also developed. Binary-coded GAs do not work on the real generator outputs, but on a bit string encoding of them. The output of the free generators is encoded in a 10-b string (an unsigned 10-b integer), which gives a resolution of 2 discrete power values in the range [ ]. These strings are concatenated to a chromosome that consists of b. The creation of the initial chromosome population is made using a random bit generator. The chromosome evaluation follows the same five-step process described in Section IV-A-1 for the real-coded GA, except from the fact that step 1 begins with the decoding of the binary chromosome to the reduced unit power output vector. The genetic evolution process incorporates all of the wellknown features of a standard binary-coded GA implementation [16] such as roulette wheel selection, crossover (multipoint), mutation, and elitism. C. Varying Operator Probabilities In order to enhance the search, varying operator probabilities are used in real and binary-coded GAs. It is well known that the parent selection method and the crossover operator lead to population convergence, while the mutation operator helps to maintain population diversity. If premature convergence or excessive diversity occurs, the search

5 202 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003 TABLE I GENERATOR OPERATING LIMITS AND QUADRATIC COST FUNCTION COEFFICIENTS TABLE II ED RESULTS FOR VARIOUS LOADS Fig. 1. Flowchart of the GA evolution process. becomes inefficient. In our implementation the crossover probability ranges from 40 to 100% per chromosome pair while the mutation probability ranges from 0 to 1% per bit or 0 to 50% per real parameter. Premature convergence is monitored by extracting statistical information from the population. When premature convergence is detected, the crossover probability is lowered by 10% while the mutation probability is increased by 0.04% per bit or 5% per real parameter. When excessive diversity occurs, the crossover probability is increased by 10% while mutation probability is lowered by 0.04% per bit or 5% per real parameter. A flowchart of the GA evolution process, which applies to both binary and real-coded GA, is depicted in Fig. 1. V. RESULTS Three test cases are considered. These test cases cover dferent aspects of the ED GA solution advantages over the traditional methods. All three test cases refer to the autonomous power system of the Greek island of Crete comprising 52 buses, 66 branches, and 18 thermal units. Specically, the first test case ignores branch power-flow limits and uses quadratic (convex) unit cost functions. The lambda iteration method is used in this case to provide the benchmark minimum operating cost. This test was designed for the comparison of the binary and real-coded GA. The second test case dfers from the first in that it incorporates branch power-flow constraints. A QP-based NC-ED provides the benchmark minimum operating cost against which the real-coded GA results are compared. Finally, the third test case applies the developed GAs to a set of generators with nonconvex cost functions. A. Economic Dispatch of Generators With Convex Cost Functions The technical limits and the quadratic cost function coefficients for the 18 units of the island of Crete are given in Table I. The maximum power output of the generators set is MW. Various tests were made with a varying percentage of the maximum power as demand. Table II summarizes the test results. The following conclusions are drawn from the results reported in Table II. Both GAs achieved near optimum costs. The real-coded GA though performed better than the binary-coded GA in all cases considered. Specically, the largest error of the real-coded GA was 0.011% while the largest binary coded GA error was 0.49%. The real-coded GA was much faster than the binary-coded GA. For 1000 generations and a population of 100 chromosomes, the execution time for the real-coded GA was one-fth of the execution time for the binary coded GA (3 s versus 15 s as shown in Table II). The tests were carried out on a Pentium II at 450 MHz with 128-Mb RAM. Fig. 2 presents various results using the real-coded GA. It is clear that not only the optimum solution is found within the first few hundred iterations but also that the population size can be signicantly decreased without a major degradation in the convergence or the accuracy of the algorithm. For example, in the second test illustrated, the

6 DAMOUSIS et al.: NETWORK-CONSTRAINED ECONOMIC DISPATCH USING REAL-CODED GENETIC ALGORITHM 203 TABLE III EXPERIMENTS WITH VARYING POPULATION SIZE. DEMAND IS 365 MW AND REAL-CODED GA RUNS FOR 1000 GENERATIONS TABLE IV EXPERIMENTS WITH VARYING POPULATION SIZE. DEMAND IS 365 MW AND REAL-CODED GA II RUNS FOR 1000 GENERATIONS Fig. 2. Effect of real-coded GA population size on augmented cost evolution. cost dference between the 100-chromosome population and the 10-chromosome population is only 0.4%. Since the execution time is proportional to the population size, it is evident that with a 20-chromosome population size the optimum solution can be found in only one-fth of the original execution time shown in Table II, that is in 0.6 s. From the previous discussion, we see that the real-coded GA performs very well in finding the optimum solution of the ED problem, while the execution time drastically decreases compared to the traditional binary coding and becomes comparable to the one of the -iteration method. B. Network-Constrained Economic Dispatch of Generators With Convex Cost Functions The 18 thermal units of Table I now inject power in the Crete island network comprising 52 buses and 66 branches. For this test case, several experiments were performed, using dferent GA population sizes. We use only the real-coded GA since it was demonstrated in the previous test case that it consistently yields better results than the binary-coded GA. The benchmark optimal cost provided by the QP-based NC-ED is $ The variation GA-II of the real-coded GA described in Section IV-A-3 is also tested. The results of these tests are summarized in Tables III and IV. A comparison between the results cited in Tables III and IV shows that real-coded GA II produces, in general (with a few exceptions), slightly better solutions than the real-coded GA with considerable improvement in execution time. On average, the real-coded GA II is 20% faster than the real-coded GA. C. Network-Constrained Economic Dispatch of Generators With Nonconvex Cost Functions The power system of the previous section is now used to test the efficiency of the proposed GA with nonconvex cost functions. Seven generators with nonconvex cost functions replace generators 2, 8 11, 13, and 15 of Table I. The incremental cost functions of the new generators as well as their minimum load operating costs are shown in Fig. 3 and Table V, respectively. Several experiments were conducted with dferent population sizes. Since the incremental cost functions of some generators are not monotonically increasing, traditional approaches cannot be used to solve the problem. Therefore, the benchmark operating cost, against which all experiments are compared, is obtained by running a real-coded GA with a population size of 500 chromosomes for generations. The experiment results are summarized in Table VI. Again, it is observed that the population size can be considerably decreased, with proportional decrease in execution time, without signicant degradation of the solution quality. The improved efficiency of the real-coded GA, together with its ability to overcome the modeling restrictions of the conventional algorithms, makes the real-coded GA algorithm particularly attractive for the solution of the ED problem.

7 204 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 1, FEBRUARY 2003 Fig. 3. Modied unit incremental cost functions (nonincreasing). TABLE V MODIFIED UNIT UPPER AND LOWER LIMITS AND MINIMUM LOAD COSTS TABLE VI TRIAL RUNS WITH VARYING POPULATION SIZE FOR 1000 GENERATIONS. DEMAND IS 365 MW USING A SET OF UNITS WITH CONVEX AND NONCONVEXCOST FUNCTIONS VI. CONCLUSIONS This paper introduces a new GA approach for the solution of the ED problem with network constraints. The coding of the generator outputs in the chromosome using floating-point numbers instead of the usual binary representation not only improved the accuracy of the algorithm but also drastically reduced the executiontime. Inthisway, thenewalgorithmretainedtheadvantagesof the GAs over the traditional ED methods but also eliminated the main disadvantage of the GAs which is the long execution time providing an efficient generic ED solution method. REFERENCES [1] M. J. Steinberg and T. H. 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Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. New York: Springer-Verlag, [25] C. Z. Janikow and Z. Michalewicz, An experimental comparison of binary and floating point representations in genetic algorithms, in Proceedings Fourth International Conference on Genetic Algorithms. San Mateo, CA: Morgan Kaufmann, 1991, pp [26] F. Herrera, M. Lozano, and J. L. Verdegay, Tuning fuzzy controllers by genetic algorithms, Int. J. Approx. Reas., vol. 12, pp , [27] O. Cordon and F. Herrera, Hybridizing genetic algorithms with sharing scheme and evolution strategies for designing approximate fuzzy rulebased systems, Fuzzy Sets Syst., vol. 118, pp , [28] M. J. Steinberg and T. H. Smith, The theory of incremental rates, Electr. Eng., vol. II, Apr Ioannis G. Damousis (S 01) was born in Thessaloniki, Greece, in April He received the Dipl.Eng. degree from the Aristotle University of Thessaloniki, Greece, in He is currently pursuing the Ph.D. degree in the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki. His research interests are in the areas of neural-network, fuzzy-logic, and GA applications in power systems. Mr. Damousis is a member of the Society of Professional Engineers of Greece.

8 DAMOUSIS et al.: NETWORK-CONSTRAINED ECONOMIC DISPATCH USING REAL-CODED GENETIC ALGORITHM 205 Anastasios G. Bakirtzis (S 77 M 79 SM 95) was born in Serres, Greece, in February He received the Dipl.Eng. degree from the National Technical University, Athens, Greece, in 1979, and the M.S. and Ph.D. degrees in electrical engineering from Georgia Institute of Technology, Atlanta, in 1981 and 1984, respectively. Currently, he is a Professor in the Electrical Engineering Department at Aristotle University of Thessaloniki, Greece. In 1984, he was a Consultant to Southern Company, Atlanta, GA. His research interests are in power system operation and control, reliability analysis, and in alternative energy sources. Petros S. Dokopoulos (M 77) was born in Athens, Greece, in September He received the Dipl.Eng. degree from the Technical University of Athens, Greece, in 1962, and the Ph.D. degree from the University of Brunswick, Germany, in Currently, he is Full Professor at the Department of Electrical Engineering at the Aristotle University of Thessaloniki, Greece, where he has been since From 1962 to 1967, he was with the Laboratory for High Voltage and Transmission at the University of Brunswick, Germany, and from 1967 to 1974, he was with the Nuclear Research Center at Julich, Germany. From 1974 to 1978, he was with the Joint European Torus, Culham, Oxforshire, U.K. His fields of interest are dielectric, power switches, power generation (conventional and fusion), transmission, distribution, and control in power systems.